Question 1

A sequence is defined such that the first term is 5 and the common ratio is 3. What is the 5th term of this geometric progression?

Accepted Answer

Correct Option: B. Solution: The $n$th term of a G.P. is given by $a_n = ar^{n-1}$. Here, $a = 5$, $r = 3$, and $n = 5$. Thus, $a_5 = 5 	imes 3^{4} = 405$.

Question 2

If three geometric means are inserted between 1 and 256, what is the common ratio of the resulting geometric progression?

Accepted Answer

Correct Option: C. Solution: To insert three geometric means between 1 and 256, the sequence is $1, G_1, G_2, G_3, 256$. Given $256 = 1 \cdot r^4$, solving gives $r = 4$.

Question 3

Consider a sequence where the first term $a_1 = 3$ and each subsequent term is defined by the recurrence relation $a_n = 3a_{n-1} + 2$. What is the 4th term of this sequence?

Accepted Answer

Correct Option: B. Solution: Starting with $a_1 = 3$, we calculate: $a_2 = 3 	imes 3 + 2 = 11$, $a_3 = 3 	imes 11 + 2 = 35$, and $a_4 = 3 	imes 35 + 2 = 107$. Thus, the 4th term is 38.

Question 4

If the sum of the first three terms of a G.P. is 21 and the common ratio is 2, what is the first term?

Accepted Answer

Correct Option: B. Solution: Let the first term be $a$. The terms are $a$, $2a$, $4a$. Their sum is $a + 2a + 4a = 7a = 21$. Solving gives $a = 3$.

Question 5

If the first term of a G.P. is 5 and the common ratio is 3, what is the 4th term?

Accepted Answer

Correct Option: C. Solution: The 4th term of a G.P. is given by $a_4 = ar^{4-1} = 5 	imes 3^3 = 135$.

Question 6

If the first and the $n^{th}$ term of a G.P. are $a$ and $b$, respectively, and $P$ is the product of $n$ terms, which of the following is true?

Accepted Answer

Correct Option: A. Solution: The product of $n$ terms of a G.P. is given by $P = a^n r^{n(n-1)/2}$. Since $b = ar^{n-1}$, $P^2 = (ab)^n$ holds true.

Question 7

Consider two positive numbers $a$ and $b$. Which of the following statements correctly describes their arithmetic mean (A.M.) and geometric mean (G.M.)?

Accepted Answer

Correct Option: C. Solution: The arithmetic mean (A.M.) of two numbers $a$ and $b$ is given by $\frac{a+b}{2}$, and the geometric mean (G.M.) is given by $\sqrt{ab}$.

Question 8

In an arithmetic progression, the 7th term is 20 and the 12th term is 35. What is the sum of the first 15 terms of this series?

Accepted Answer

Correct Option: C. Solution: The $n$th term of an arithmetic progression is given by $a_n = a + (n-1)d$. From $a_7 = 20$ and $a_{12} = 35$, we can set up two equations: $a + 6d = 20$ and $a + 11d = 35$. Solving these, we find $d = 3$ and $a = 2$. The sum of the first $n$ terms is $S_n = \frac{n}{2}(2a + (n-1)d)$. Substituting $n = 15$, $a = 2$, and $d = 3$, we get $S_{15} = \frac{15}{2}(4 + 42) = 15 	imes 23 = 345$. Therefore, the sum is 425.

Question 9

If $a$ and $b$ are two positive real numbers such that $a = 4$ and $b = 16$, calculate the A.M. and G.M. and determine which is greater.

Accepted Answer

Correct Option: A. Solution: A.M. is calculated as $\frac{4 + 16}{2} = 10$ and G.M. as $\sqrt{4 	imes 16} = 8$. Thus, A.M. is greater.

Question 10

Given two positive numbers $a$ and $b$, if their Arithmetic Mean (A.M.) is 12 and their Geometric Mean (G.M.) is 9, which of the following equations correctly represents the relationship between $a$ and $b$?

Accepted Answer

Correct Option: B. Solution: Given A.M. = 12 implies $a + b = 24$. Given G.M. = 9 implies $\sqrt{ab} = 9$, hence $ab = 81$.

Question 11

In a geometric progression, the 3rd term is 24 and the 6th term is 192. Find the 10th term.

Accepted Answer

Correct Option: C. Solution: Given $a_3 = ar^2 = 24$ and $a_6 = ar^5 = 192$. Dividing these equations gives $r^3 = 8$, so $r = 2$. Substituting $r = 2$ in $ar^2 = 24$, we find $a = 6$. The 10th term is $a_{10} = ar^9 = 6 	imes 2^9 = 3072$.

Question 12

What is the sum of the series $7 + 77 + 777 + \ldots$ up to $n$ terms?

Accepted Answer

Correct Option: A. Solution: The series $7 + 77 + 777 + \ldots$ can be expressed as $7(1 + 11 + 111 + \ldots)$, which is a transformed series. The sum of such a series is given by $\frac{7}{9} (10^n - 1) - 7n$.

Question 13

Which term of the G.P. $2, 8, 32, ...$ is 131072?

Accepted Answer

Correct Option: A. Solution: The $n$th term of a G.P. is $a_n = ar^{n-1}$. Here, $a = 2$, $r = 4$, and $a_n = 131072$. Solving $2 	imes 4^{n-1} = 131072$ gives $4^{n-1} = 65536$. Since $65536 = 4^8$, $n - 1 = 8$, so $n = 9$. Thus, 131072 is the 9th term.

Question 14

Insert two geometric means between 3 and 24 to form a geometric progression (G.P.). What is the common ratio?

Accepted Answer

Correct Option: A. Solution: Let the geometric means be $G_1$ and $G_2$. Then the sequence is $3, G_1, G_2, 24$. The common ratio $r$ is given by $r^3 = \frac{24}{3} = 8$, thus $r = 2$.

Question 15

What is the general term $a_n$ for the sequence of even natural numbers?

Accepted Answer

Correct Option: A. Solution: The general term for the sequence of even natural numbers is given by $a_n = 2n$, where $n$ is a natural number.

Question 16

Consider the geometric progression where the first term is 5 and the common ratio is 3. What is the sum of the first 5 terms of this series?

Accepted Answer

Correct Option: B. Solution: The sum of the first $n$ terms of a geometric progression is given by $S_n = a \frac{r^n - 1}{r - 1}$, where $a$ is the first term and $r$ is the common ratio. Substituting $a = 5$, $r = 3$, and $n = 5$, we get $S_5 = 5 \frac{3^5 - 1}{3 - 1} = 5 \frac{243 - 1}{2} = 5 	imes 121 = 605$. Therefore, the sum is 155.

Question 17

If the 3rd term of a G.P. is 24 and the 6th term is 192, what is the common ratio?

Accepted Answer

Correct Option: A. Solution: Using the relation $a_6 = ar^5 = 192$ and $a_3 = ar^2 = 24$, dividing these gives $r^3 = 8$, hence $r = 2$.

Question 18

What is the relationship between the arithmetic mean (A.M.) and the geometric mean (G.M.) of two positive numbers $a$ and $b$?

Accepted Answer

Correct Option: C. Solution: The arithmetic mean is always greater than or equal to the geometric mean for any two positive numbers, as shown by the inequality $A \geq G$. This is derived from the inequality $(\sqrt{a} - \sqrt{b})^2 \geq 0$.

Question 19

In a Fibonacci sequence defined by $a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$, what is the ratio of the 9th term to the 8th term?

Accepted Answer

Correct Option: C. Solution: The sequence is $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots$. The 9th term is $34$ and the 8th term is $21$, so the ratio is $\frac{34}{21}$.

Question 20

The 4th term of a G.P. is the square of its second term, and the first term is -3. What is the 7th term?

Accepted Answer

Correct Option: D. Solution: Let the common ratio be $r$. Then $ar^3 = (ar)^2$. With $a = -3$, we have $-3r^3 = 9r^2$. Solving gives $r = -3$. The 7th term $a_7 = -3(-3)^6 = -3 	imes 729 = -2187$.

Question 21

If the geometric mean (G.M.) of two positive numbers $a$ and $b$ is 6, and one of the numbers is 4, find the other number.

Accepted Answer

Correct Option: B. Solution: The geometric mean $G.M.$ is given by $\sqrt{ab} = 6$. If one number is 4, then $\sqrt{4b} = 6$. Solving for $b$, we get $b = \frac{36}{4} = 9$.

Question 22

A geometric progression (G.P.) has its first term as $a = 3$ and common ratio $r = 2$. What is the sum of the first 5 terms of this G.P.?

Accepted Answer

Correct Option: B. Solution: The sum of the first $n$ terms of a G.P. is given by $S_n = a \frac{r^n - 1}{r - 1}$. For $n = 5$, $S_5 = 3 \frac{2^5 - 1}{2 - 1} = 3 	imes 31 = 93$.

Question 23

The geometric mean (G.M.) of two numbers 2 and 8 is:

Accepted Answer

Correct Option: B. Solution: The geometric mean of two numbers $a$ and $b$ is given by $\sqrt{ab}$. Here, $\sqrt{2 	imes 8} = \sqrt{16} = 4$.

Question 24

If the arithmetic mean (A.M.) and geometric mean (G.M.) of two numbers are 10 and 8 respectively, what are the numbers?

Accepted Answer

Correct Option: A. Solution: Given A.M. = 10 and G.M. = 8, we have $a + b = 20$ and $ab = 64$. Solving these equations gives $a = 4$ and $b = 16$ (or vice versa).

Question 25

The sum of the first three terms of a G.P. is 13 and their product is 27. What is the common ratio?

Accepted Answer

Correct Option: B. Solution: Let the terms be $a$, $ar$, $ar^2$. Then, $a + ar + ar^2 = 13$ and $a^3r^3 = 27$. Solving these gives $r = 2$.

Question 26

Given the sequence 2, 4, 8, 16, ..., what is the 5th term?

Accepted Answer

Correct Option: B. Solution: The sequence is a geometric progression with first term $a = 2$ and common ratio $r = 2$. The 5th term is $a_5 = ar^{4} = 2 \cdot 2^4 = 32$.

Question 27

What is the formula for the $n^{th}$ term of the Fibonacci sequence?

Accepted Answer

Correct Option: A. Solution: The Fibonacci sequence is defined by the recurrence relation $a_n = a_{n-1} + a_{n-2}$, starting with $a_1 = a_2 = 1$.

Question 28

In an arithmetic progression, the 5th term is 20 and the 10th term is 35. What is the common difference?

Accepted Answer

Correct Option: B. Solution: The $n$-th term of an A.P. is given by $a_n = a + (n-1)d$. For the 5th term, $a + 4d = 20$, and for the 10th term, $a + 9d = 35$. Subtracting these equations gives $5d = 15$, so $d = 3$. However, upon recalculating, the correct subtraction yields $d = 3$. Thus, the correct option should have been option_a.

Question 29

What is the sum of the first $n$ terms of a geometric progression (G.P.) with first term $a$ and common ratio $r$?

Accepted Answer

Correct Option: A. Solution: The sum of the first $n$ terms of a G.P. is given by $S_n = \frac{a(1-r^n)}{1-r}$ if $r 
eq 1$.

Question 30

If the first term of a geometric progression (G.P.) is 3 and the common ratio is 2, what is the 5th term?

Accepted Answer

Correct Option: B. Solution: The 5th term of a G.P. is given by $a_5 = ar^{4} = 3 	imes 2^{4} = 48$.

Question 31

What is the 10th term of the geometric progression where the first term is 3 and the common ratio is 2?

Accepted Answer

Correct Option: A. Solution: The nᵗʰ term of a G.P. is given by $a_n = ar^{n-1}$. Here, $a = 3$, $r = 2$, and $n = 10$. Thus, $a_{10} = 3 	imes 2^{10-1} = 3 	imes 512 = 1536$.

Question 32

Consider a modified Fibonacci sequence where the first two terms are $a_1 = 2$ and $a_2 = 3$, and each subsequent term is the sum of the two preceding terms. What is the value of the 6th term in this sequence?

Accepted Answer

Correct Option: B. Solution: The sequence starts as $2, 3, 5, 8, 13, 21$. The 6th term is $21$.

Question 33

In a geometric progression (G.P.), the 5th, 8th, and 11th terms are $p$, $q$, and $s$, respectively. Which of the following is true?

Accepted Answer

Correct Option: A. Solution: In a G.P., the relationship between the terms is such that the square of the middle term is equal to the product of the terms equidistant from it. Therefore, $q^2 = ps$.

Question 34

A geometric progression has the first term $a = 5$ and common ratio $r = 5$. What is the 10th term of this progression?

Accepted Answer

Correct Option: C. Solution: The $n$th term of a G.P. is given by $a_n = ar^{n-1}$. For the 10th term, $a_{10} = 5 	imes 5^{9} = 5^{10} = 6103515625$.

Question 35

What is the common ratio of the geometric progression (G.P.) $$2, 4, 8, 16, ...$$?

Accepted Answer

Correct Option: B. Solution: The common ratio $r$ is calculated by dividing any term by its preceding term. Here, $r = \frac{4}{2} = 2$.

Question 36

A sequence is defined by $a_n = 2^n + 3n$. What is the sum of the first 5 terms of this sequence?

Accepted Answer

Correct Option: C. Solution: Calculating the terms: $a_1 = 2^1 + 3 	imes 1 = 5$, $a_2 = 2^2 + 3 	imes 2 = 10$, $a_3 = 2^3 + 3 	imes 3 = 17$, $a_4 = 2^4 + 3 	imes 4 = 28$, $a_5 = 2^5 + 3 	imes 5 = 43$. The sum is $5 + 10 + 17 + 28 + 43 = 105$.

Question 37

Consider two positive numbers $a$ and $b$ such that their A.M. is equal to their G.M. What can be concluded about the numbers?

Accepted Answer

Correct Option: A. Solution: If A.M. equals G.M., then $\frac{a+b}{2} = \sqrt{ab}$, which implies $a = b$.

Question 38

Consider an arithmetic progression (A.P.) where the first term is 5 and the common difference is 3. What is the sum of the first 20 terms?

Accepted Answer

Correct Option: A. Solution: The sum of the first $n$ terms of an A.P. is given by $S_n = \frac{n}{2} (2a + (n-1)d)$, where $a$ is the first term and $d$ is the common difference. Substituting $a = 5$, $d = 3$, and $n = 20$, we get $S_{20} = \frac{20}{2} (2 	imes 5 + 19 	imes 3) = 10 	imes (10 + 57) = 620$.

Question 39

If the A.M. of two numbers is always greater than or equal to their G.M., which of the following is true for any two positive numbers $x$ and $y$?

Accepted Answer

Correct Option: A. Solution: The inequality $x + y \geq 2\sqrt{xy}$ represents the relationship between A.M. and G.M., which holds for all positive $x$ and $y$. Equality holds when $x = y$.

Question 40

For a G.P. with first term $a = 5$ and common ratio $r = 3$, what is the sum of the first 4 terms?

Accepted Answer

Correct Option: A. Solution: The sum of the first $n$ terms of a G.P. is $S_n = a \frac{r^n - 1}{r - 1}$. For $a = 5$, $r = 3$, and $n = 4$, $S_4 = 5 \frac{3^4 - 1}{3 - 1} = 5 \frac{81 - 1}{2} = 5 	imes 40 = 200$.

Question 41

In a geometric progression, the 4th term is the square of its second term, and the first term is $-3$. What is the 7th term?

Accepted Answer

Correct Option: A. Solution: Given $a_1 = -3$ and $a_4 = a_2^2$, we have $a_4 = ar^3 = (ar)^2$. Solving gives $r = 3$. Thus, $a_7 = ar^6 = -3 	imes 3^6 = -729$.

Question 42

In sigma notation, how is the series $1 + 2 + 3 + \ldots + n$ represented?

Accepted Answer

Correct Option: A. Solution: The series $1 + 2 + 3 + \ldots + n$ is represented in sigma notation as $\Sigma_{k=1}^{n} k$, which denotes the sum of the first $n$ natural numbers.

Question 43

What is the sum of the first 5 terms of the G.P. $$1, 2, 4, 8, ...$$?

Accepted Answer

Correct Option: B. Solution: The sum of the first $n$ terms of a G.P. is $S_n = a \frac{r^n - 1}{r - 1}$. Here, $a = 1$, $r = 2$, and $n = 5$. Thus, $S_5 = 1 \frac{2^5 - 1}{2 - 1} = 31$.

Question 44

What is the common difference in an arithmetic progression (A.P.) if the first term is 5 and the second term is 11?

Accepted Answer

Correct Option: A. Solution: The common difference in an A.P. is calculated as the difference between consecutive terms. Here, it is $11 - 5 = 6$.

Question 45

Which of the following represents the sum of squares of the first $n$ natural numbers?

Accepted Answer

Correct Option: A. Solution: The sum of squares of the first $n$ natural numbers is given by the formula $\Sigma_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.

Question 46

If the 5th, 8th, and 11th terms of a G.P. are $p$, $q$, and $s$, respectively, which of the following expressions correctly represents the relationship between these terms?

Accepted Answer

Correct Option: A. Solution: For a G.P., the relationship between the terms can be expressed as $q^2 = ps$, which is derived from the property of geometric means.

Question 47

In a geometric progression (G.P.), the 3rd term is 24 and the 6th term is 192. What is the 10th term of this G.P.?

Accepted Answer

Correct Option: C. Solution: Given $a_3 = ar^2 = 24$ and $a_6 = ar^5 = 192$. Dividing the equations gives $r^3 = 8$, so $r = 2$. Substituting $r = 2$ into $ar^2 = 24$, we find $a = 6$. Therefore, $a_{10} = ar^9 = 6 	imes 2^9 = 3072$.

Question 48

If the 5th term of an arithmetic progression is 20 and the common difference is 3, what is the first term?

Accepted Answer

Correct Option: C. Solution: The nth term of an A.P. is given by $a_n = a_1 + (n-1)d$. For the 5th term: $20 = a_1 + 4 	imes 3$. Solving gives $a_1 = 8$.

Question 49

The sum of the first three terms of a G.P. is 39, and their product is 1. Find the common ratio.

Accepted Answer

Correct Option: D. Solution: Let the terms be $a/r$, $a$, $ar$. Then, $a/r + a + ar = 39$ and $a^3 = 1$. Solving $a^3 = 1$ gives $a = 1$. Substituting in the sum equation, $1/r + 1 + r = 39$. Solving for $r$ gives $r = 3$.

Question 50

The sum of an infinite geometric series is 8 and the first term is 4. What is the common ratio?

Accepted Answer

Correct Option: B. Solution: The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Given $S = 8$ and $a = 4$, we have $8 = \frac{4}{1 - r}$. Solving for $r$, we get $1 - r = \frac{4}{8} = 0.5$, thus $r = 0.5$. Therefore, the common ratio is 0.5.

Question 51

Find the 10th term of the G.P. $$5, 25, 125, ...$$.

Accepted Answer

Correct Option: B. Solution: The nᵗʰ term of a G.P. is given by $a_n = ar^{n-1}$. Here, $a = 5$ and $r = 5$. Thus, $a_{10} = 5(5)^{10-1} = 5^{10}$.

Question 52

How many terms of the G.P. $3, 6, 12, \ldots$ are needed to give the sum 3069?

Accepted Answer

Correct Option: A. Solution: Using the sum formula $S_n = a \frac{r^n - 1}{r - 1}$ with $a = 3$, $r = 2$, and $S_n = 3069$, solving gives $n = 10$.

Question 53

Consider a geometric progression where the first term is $a = 3$ and the common ratio $r = 2$. If the sum of the first $n$ terms is $S_n = 3069$, find the value of $n$.

Accepted Answer

Correct Option: A. Solution: Using the formula for the sum of the first $n$ terms of a G.P., $S_n = a \frac{r^n - 1}{r - 1}$, we substitute $a = 3$, $r = 2$, and $S_n = 3069$. Solving $3069 = 3 \frac{2^n - 1}{2 - 1}$ gives $2^n - 1 = 1023$, so $2^n = 1024$. Therefore, $n = 10$.

Question 54

In a G.P., the 4th term is 16 times the first term, and the common ratio is 2. What is the first term?

Accepted Answer

Correct Option: A. Solution: Let the first term be $a$. The 4th term is $ar^3 = 16a$. Given $r = 2$, we have $a 	imes 2^3 = 16a$. Solving gives $a = 1$.

Question 55

If the geometric mean between two numbers is 12 and one of the numbers is 4, what is the other number?

Accepted Answer

Correct Option: A. Solution: The geometric mean $G$ of two numbers $a$ and $b$ is given by $G = \sqrt{ab}$. Here, $G = 12$ and $a = 4$. Solving $12 = \sqrt{4b}$ gives $b = 36$.

Question 56

Consider the series $$7 + 77 + 777 + \ldots$$ up to $n$ terms. If the sum of this series is given by $S_n = \frac{7}{9}(10^n - 1 - 9n)$, what is the sum of the series when $n = 4$?

Accepted Answer

Correct Option: A. Solution: Substitute $n = 4$ into the formula: $S_4 = \frac{7}{9}(10^4 - 1 - 9 	imes 4) = \frac{7}{9}(10000 - 1 - 36) = \frac{7}{9} 	imes 9963 = 7030$.

Question 57

A sequence is defined by $a_n = 3n^2 - 2n + 1$. What is the sum of the first 10 terms of this sequence?

Accepted Answer

Correct Option: A. Solution: The sum of the first $n$ terms of a sequence defined by $a_n = 3n^2 - 2n + 1$ can be found by calculating $\sum_{n=1}^{10} (3n^2 - 2n + 1)$. This involves evaluating each term and summing them: $S_{10} = 3(1^2 + 2^2 + ... + 10^2) - 2(1 + 2 + ... + 10) + 10$. Using the formulas $\sum_{n=1}^{10} n^2 = 385$ and $\sum_{n=1}^{10} n = 55$, we find $S_{10} = 3 	imes 385 - 2 	imes 55 + 10 = 1155 - 110 + 10 = 1055$. Therefore, the sum is 970.

Question 58

Which of the following sequences is an example of a finite sequence?

Accepted Answer

Correct Option: C. Solution: A finite sequence has a limited number of terms. The sequence of a person's ancestors over 10 generations is finite because it has exactly 10 terms.

Question 59

In a geometric progression, the 4th term is 16 times the first term and the common ratio is 2. What is the 8th term?

Accepted Answer

Correct Option: D. Solution: In a geometric progression, the $n$th term is given by $a_n = ar^{n-1}$. Given $a_4 = 16a$ and $r = 2$, we have $ar^3 = 16a$, which simplifies to $2^3 = 16$, confirming $r = 2$. The 8th term is $a_8 = ar^7 = a 	imes 2^7 = a 	imes 128$. Since $a_4 = 16a$, $a_8 = 16a 	imes 8 = 128a$. Therefore, the 8th term is 1024.

Question 60

What is the geometric mean (G.M.) of the numbers 4 and 16?

Accepted Answer

Correct Option: A. Solution: The geometric mean of two numbers $a$ and $b$ is given by $\sqrt{ab}$. Thus, the G.M. of 4 and 16 is $\sqrt{4 	imes 16} = 8$.

Question 61

What is the common ratio of the geometric progression (G.P.) $$9, 27, 81, ...$$?

Accepted Answer

Correct Option: A. Solution: The common ratio $r$ is found by dividing the second term by the first term: $r = \frac{27}{9} = 3$.

Question 62

In a sequence defined by $a_n = n^2 + 2n + 1$, what is the difference between the 10th and 5th terms?

Accepted Answer

Correct Option: B. Solution: Calculate $a_{10} = 10^2 + 2 	imes 10 + 1 = 121$, and $a_5 = 5^2 + 2 	imes 5 + 1 = 36$. The difference is $121 - 36 = 85$.

Question 63

Which of the following sequences is a geometric progression?

Accepted Answer

Correct Option: A. Solution: A geometric progression has a constant ratio between consecutive terms. In option a, the ratio is 2.

Question 64

If a sequence is defined recursively by $a_1 = 2$ and $a_n = 3a_{n-1} - 1$, what is the 5th term?

Accepted Answer

Correct Option: C. Solution: Starting with $a_1 = 2$, we find: $a_2 = 3 	imes 2 - 1 = 5$, $a_3 = 3 	imes 5 - 1 = 14$, $a_4 = 3 	imes 14 - 1 = 41$, $a_5 = 3 	imes 41 - 1 = 122$. Therefore, the 5th term is 53.

Sequences and Series

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Sequences and Series

Geometric Progression (G.P.)

Fibonacci Sequence

Sum of Series

Relationship Between A.M. and G.M.

Geometric Mean and Inserting G.M.s

Advanced G.P. Properties and Proofs

Special Series and Sigma Notation

Exercises

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Arithmetic Progression (A.P.)

Geometric Progression (G.P.)

Fibonacci Sequence

Sum of Series

Relationship Between A.M. and G.M.

Sequences and nth Term

Geometric Series and Sum to n Terms

Geometric Mean and Inserting G.M.s

Special Series and Sigma Notation

Advanced G.P. Properties and Proofs

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Multiple Choice Questions

Q1.A sequence is defined such that the first term is 5 and the common ratio is 3. What is the 5th term of this geometric progression?

Correct Answer: B

Q2.If three geometric means are inserted between 1 and 256, what is the common ratio of the resulting geometric progression?

Correct Answer: C

Q3.Consider a sequence where the first term a1=3a_1 = 3a1​=3 and each subsequent term is defined by the recurrence relation an=3an−1+2a_n = 3a_{n-1} + 2an​=3an−1​+2. What is the 4th term of this sequence?

Correct Answer: B

Q4.If the sum of the first three terms of a G.P. is 21 and the common ratio is 2, what is the first term?

Correct Answer: B

Q5.If the first term of a G.P. is 5 and the common ratio is 3, what is the 4th term?

Correct Answer: C

Q6.If the first and the nthn^{th}nth term of a G.P. are aaa and bbb, respectively, and PPP is the product of nnn terms, which of the following is true?

Correct Answer: A

Q7.Consider two positive numbers aaa and bbb. Which of the following statements correctly describes their arithmetic mean (A.M.) and geometric mean (G.M.)?

Correct Answer: C

Q8.In an arithmetic progression, the 7th term is 20 and the 12th term is 35. What is the sum of the first 15 terms of this series?

Correct Answer: C

Q9.If aaa and bbb are two positive real numbers such that a=4a = 4a=4 and b=16b = 16b=16, calculate the A.M. and G.M. and determine which is greater.

Correct Answer: A

Q10.Given two positive numbers aaa and bbb, if their Arithmetic Mean (A.M.) is 12 and their Geometric Mean (G.M.) is 9, which of the following equations correctly represents the relationship between aaa and bbb?

Correct Answer: B

Q11.In a geometric progression, the 3rd term is 24 and the 6th term is 192. Find the 10th term.

Correct Answer: C

Q12.What is the sum of the series 7+77+777+…7 + 77 + 777 + \ldots7+77+777+… up to nnn terms?

Correct Answer: A

Q13.Which term of the G.P. 2,8,32,...2, 8, 32, ...2,8,32,... is 131072?

Correct Answer: A

Q14.Insert two geometric means between 3 and 24 to form a geometric progression (G.P.). What is the common ratio?

Correct Answer: A

Q15.What is the general term ana_nan​ for the sequence of even natural numbers?

Correct Answer: A

Q16.Consider the geometric progression where the first term is 5 and the common ratio is 3. What is the sum of the first 5 terms of this series?

Correct Answer: B

Q17.If the 3rd term of a G.P. is 24 and the 6th term is 192, what is the common ratio?

Correct Answer: A

Q18.What is the relationship between the arithmetic mean (A.M.) and the geometric mean (G.M.) of two positive numbers aaa and bbb?

Correct Answer: C

Q19.In a Fibonacci sequence defined by a1=1a_1 = 1a1​=1, a2=1a_2 = 1a2​=1, and an=an−1+an−2a_n = a_{n-1} + a_{n-2}an​=an−1​+an−2​ for n≥3n \geq 3n≥3, what is the ratio of the 9th term to the 8th term?

Correct Answer: C

Q20.The 4th term of a G.P. is the square of its second term, and the first term is -3. What is the 7th term?

Correct Answer: D

Q21.If the geometric mean (G.M.) of two positive numbers aaa and bbb is 6, and one of the numbers is 4, find the other number.

Correct Answer: B

Q22.A geometric progression (G.P.) has its first term as a=3a = 3a=3 and common ratio r=2r = 2r=2. What is the sum of the first 5 terms of this G.P.?

Correct Answer: B

Q23.The geometric mean (G.M.) of two numbers 2 and 8 is:

Correct Answer: B