Geometric Progression (G.P.)

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JEE Main 2016

LEVELJEE Main

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

Three positive numbers form an increasing G. P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is:

(A)

2 - 3

(B)

2 + 3

(C)

2 + 3

(D)

3 + 2

JEE Main 2013

LEVELBoard

The sum of first $20$ terms of the sequence $0.7, 0.77, 0.777, \dots$ is

(A)

\frac{7}{81} (179 - 1 0^{- 20})

(B)

\frac{7}{9} (99 - 1 0^{- 20})

(C)

\frac{7}{81} (179 + 1 0^{- 20})

(D)

\frac{7}{9} (99 + 1 0^{- 20})

JEE Main 2008

LEVELBoard

The first two terms of a geometric progression add up to $12$ . the sum of the third and the fourth terms is $48$ . If the terms of the geometric progression are alternately positive and negative, then the first term is

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals

(A)

5

(B)

\frac{1}{2} (5 - 1)

(C)

\frac{1}{2} (1 - 5)

(D)

\frac{5}{2}

JEE Main 2003

LEVELJEE Main

If $x_{1}, x_{2}, x_{3}$ and $y_{1}, y_{2}, y_{3}$ are both in G.P. with the same common ratio, then the points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$

(A)

are vertices of a triangle

(B)

lie on a straight line

Fifth term of a GP is $2$ , then the product of its $9$ terms is

Sum of infinite number of terms of GP is $20$ and sum of their square is $100$ . The common ratio of GP is

Suppose four distinct positive numbers $a_{1}, a_{2}, a_{3}, a_{4}$ are in G.P. Let $b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3}$ and $b_{4} = b_{3} + a_{4}$ . STATEMENT - 1 : The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are neither in A.P. nor in G.P. and STATEMENT - 2 : The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are in H.P.

(A)

STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1

(B)

STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1

(C)

STATEMENT - 1 is True, STATEMENT - 2 is False

(D)

STATEMENT - 1 is False, STATEMENT - 2 is True

JEE Advanced 2004

LEVELBoard

An infinite G.P. has first term ' $x$ ' and sum ' $5$ ', then $x$ belongs to

(A)

x < - 10

(B)

- 10 < x < 0

(C)

0 < x < 10

(D)

x > 10

JEE Advanced 2000

LEVELJEE Main

Consider an infinite geometric series with first term $a$ and common ratio $r$ . If its sum is $4$ and the second term is $3/4$ , then

(A)

a = 4/7, r = 3/7

(B)

a = 2, r = 3/8

(C)

a = 3/2, r = 1/2

(D)

a = 3, r = 1/4

JEE Advanced 1999

LEVELJEE Main

Let $S_{1}, S_{2}, \dots$ be squares such that for each $n \geq 1$ , the length of a side of $S_{n}$ equals the length of a diagonal of $S_{n + 1}$ . If the length of a side of $S_{1}$ is 10 cm, then for which of the following values of $n$ is the area of $S_{n}$ less than 1 sq. cm?

* Multiple Correct Options

If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^{2} + b^{2} + c^{2}) p^{2} - 2 (ab + b c + c d) p + (b^{2} + c^{2} + d^{2}) \leq 0$ then $a, b, c, d$

ab = c d

(E)

satisfy none of these

JEE Advanced 1983

LEVELBoard

The rational number, which equals the number $2.357$ with recurring decimal is

(A)

\frac{2355}{1001}

(B)

\frac{2379}{997}

(C)

\frac{2355}{999}

If $x, y$ and $z$ are $p$ th, $q$ th and $r$ th terms respectively of an A.P. and also of a G.P., then $x^{y - z} y^{z - x} z^{x - y}$ is equal to :

The third term of a geometric progression is $4$ . The product of the first five terms is

(A)

4^{3}

(B)

4^{5}

(C)

4^{4}

If $a_{n} = \frac{3}{4} - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} + \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $b_{n} = 1 - a_{n}$ , then find the least natural number $n_{0}$ such that $b_{n} > a_{n}$ for all $n \geq n_{0}$ .

JEE Advanced 1992

LEVELJEE Main

Find the natural number 'a' for which $\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1)$ , where the function 'f' satisfies the relation $f (x + y) = f (x) f (y)$ for all natural numbers $x, y$ and further $f (1) = 2$ .

JEE Advanced 1991

LEVELJEE Main

If $S_{1}, S_{2}, S_{3}, \dots, S_{n}$ are the sums of infinite geometric series whose first terms are $1, 2, 3, \dots, n$ and whose common ratios are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots, \frac{1}{n + 1}$ respectively, then find the values of $S_{1}^{2} + S_{2}^{2} + S_{3}^{2} + \dots + S_{2 n - 1}^{2}$ .

JEE Advanced 1983

LEVELJEE Main

Find three numbers $a, b, c$ , between $2$ and $18$ such that (i) their sum is $25$ (ii) the numbers $2, a, b$ sare consecutive terms of an A.P. and (iii) the numbers $b, c, 18$ are consecutive terms of a G.P.

JEE Advanced 1982

LEVELJEE Main

Does there exist a geometric progression containing $27, 8$ and $12$ as three of its terms ? If it exits, how many such progressions are possible ?

JEE Advanced 2001

LEVELJEE Main

Let $a, b, c$ be positive real numbers such that $b^{2} - 4 a c > 0$ and let $α_{1} = c$ . Prove by induction that $α_{n + 1} = \frac{a α _{n}^{2}}{b ^{2} - 2 a ( α _{1} + α _{2} + .... + α _{n} )}$ is well-defined and $α_{n + 1} < \frac{α _{n}}{2}$ for all $n = 1, 2, ...$ (Here, 'well-defined' means that the denominator in the expression for $α_{n + 1}$ is not zero.)

JEE Main 2018 (15 April Shift 1)

LEVELBoard

If $b$ is the first term of an infinite G.P. whose sum is five, then $b$ lies in the interval :

(A)

(- \infty, - 10]

(B)

(- 10, 0)

(C)

(0, 10)

(D)

[10, \infty)

JEE Main 2018 (15 April Evening)

LEVELJEE Main

Let, $A_{n} = (\frac{3}{4}) - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} - \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $B_{n} = 1 - A_{n}$ . Then, the least odd natural number p, so that $B_{n} > A_{n}$ for all $n \geq p$ , is :-

JEE Main 2018 (15 April Evening)

LEVELJEE Main

If a, b, c are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. such that $a < b < c$ and $a + b + c = \frac{3}{4}$ , then the value of a is :-

(A)

\frac{1}{4} - \frac{1}{2}

(B)

\frac{1}{4} - \frac{1}{3 2}

(C)

\frac{1}{4} - \frac{1}{4 2}

(D)

\frac{1}{4} - \frac{1}{2 2}

If the 2nd, 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is:

Three positive numbers form an increasing G. P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of first 20 terms of the sequence 0.7,0.77,0.777,… is

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x1​,x2​,x3​ and y1​,y2​,y3​ are both in G.P. with the same common ratio, then the points (x1​,y1​),(x2​,y2​) and (x3​,y3​)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Fifth term of a GP is 2, then the product of its 9 terms is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } An infinite G.P. has first term 'x' and sum '5', then x belongs to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is 3/4, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c,d and p are distinct real numbers such that (a2+b2+c2)p2−2(ab+bc+cd)p+(b2+c2+d2)≤0 then a,b,c,d

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The rational number, which equals the number 2.357 with recurring decimal is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x,y and z are pth, qth and rth terms respectively of an A.P. and also of a G.P., then xy−zyz−xzx−y is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The third term of a geometric progression is 4. The product of the first five terms is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If an​=43​−(43​)2+(43​)3+⋯+(−1)n−1(43​)n and bn​=1−an​, then find the least natural number n0​ such that bn​>an​ for all n≥n0​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Does there exist a geometric progression containing 27,8 and 12 as three of its terms ? If it exits, how many such progressions are possible ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let, An​=(43​)−(43​)2+(43​)3−⋯+(−1)n−1(43​)n and Bn​=1−An​. Then, the least odd natural number p, so that Bn​>An​ for all n≥p, is :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a, b, c are in A.P. and a2,b2,c2 are in G.P. such that a<b<c and a+b+c=43​, then the value of a is :-

The sum of first $20$ terms of the sequence $0.7, 0.77, 0.777, \dots$ is

If $x_{1}, x_{2}, x_{3}$ and $y_{1}, y_{2}, y_{3}$ are both in G.P. with the same common ratio, then the points $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$

Fifth term of a GP is $2$ , then the product of its $9$ terms is

Sum of infinite number of terms of GP is $20$ and sum of their square is $100$ . The common ratio of GP is

An infinite G.P. has first term ' $x$ ' and sum ' $5$ ', then $x$ belongs to

Consider an infinite geometric series with first term $a$ and common ratio $r$ . If its sum is $4$ and the second term is $3/4$ , then

If $a, b, c, d$ and $p$ are distinct real numbers such that $(a^{2} + b^{2} + c^{2}) p^{2} - 2 (ab + b c + c d) p + (b^{2} + c^{2} + d^{2}) \leq 0$ then $a, b, c, d$

The rational number, which equals the number $2.357$ with recurring decimal is

If $x, y$ and $z$ are $p$ th, $q$ th and $r$ th terms respectively of an A.P. and also of a G.P., then $x^{y - z} y^{z - x} z^{x - y}$ is equal to :

The third term of a geometric progression is $4$ . The product of the first five terms is

If $a_{n} = \frac{3}{4} - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} + \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $b_{n} = 1 - a_{n}$ , then find the least natural number $n_{0}$ such that $b_{n} > a_{n}$ for all $n \geq n_{0}$ .

Does there exist a geometric progression containing $27, 8$ and $12$ as three of its terms ? If it exits, how many such progressions are possible ?

If $b$ is the first term of an infinite G.P. whose sum is five, then $b$ lies in the interval :

Let, $A_{n} = (\frac{3}{4}) - (\frac{3}{4})^{2} + (\frac{3}{4})^{3} - \dots + (- 1)^{n - 1} (\frac{3}{4})^{n}$ and $B_{n} = 1 - A_{n}$ . Then, the least odd natural number p, so that $B_{n} > A_{n}$ for all $n \geq p$ , is :-

If a, b, c are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. such that $a < b < c$ and $a + b + c = \frac{3}{4}$ , then the value of a is :-