Arithmetic & Geometric Progression Questions with Solutions

42 questions13 easy21 medium8 hard

An arithmetic progression (AP) is a sequence where each term differs from the previous one by a constant amount, the common difference. A geometric progression (GP) is a sequence where each term is the previous one multiplied by a constant, the common ratio. This topic covers the nth term and sum of n terms of an AP and GP, the sum of an infinite GP, arithmetic and geometric means, and word problems. Key formulas are the AP nth term a + (n - 1)d, the AP sum n/2 [2a + (n - 1)d], the GP nth term ar^(n-1), and the infinite GP sum a/(1 - r) for |r| < 1. A useful trick is to choose symmetric terms, such as a - d, a, a + d for three terms of an AP, which simplifies many questions. Practise the AP and GP questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
$x$ is the G.M. and $y$ is the H.M. of two numbers. If $\frac{x}{y} = \frac{5}{4}$ , then the numbers are in the ratio
A. $1 : 4$ B. $4 : 5$ C. $3 : 2$ D. $2 : 1$
hardAsked in MAH MCA CET 2023View solution ›
Q2.
Sum of squares of odd positive integers from 1 to 100 belongs to
A. 150000 to 250000B. 250000 to 350000C. 350000 to 450000D. more than 500000
mediumAsked in MAH MCA CET 2023View solution ›
Q3.
The roots of the quadratic equation $3 x^{2} - p x + q = 0$ are the $10^{t h}$ and $11^{t h}$ terms of an arithmetic progression with common difference $\frac{3}{2}$ . If the sum of the first 11 terms of this arithmetic progression is 88, then $q - 2 p$ is:
A. 464B. 474C. 484D. 494
mediumAsked in NIMCET 2026View solution ›
Q4.
Which of the following is a value of $n$ if $(\sum_{k = 1}^{n} (- 1)^{k - 1} k)^{2} - \sum_{k = 1}^{n} (- 1)^{k - 1} k^{2} + 2450 = 0$ ?
A. 98B. 99C. 100D. 101
hardAsked in NIMCET 2026View solution ›
Q5.
The arithmetic mean of two numbers a and b is 5, and the harmonic mean is 3.2. Find the numbers a and b.
A. a = 3, b = 7B. a = 4, b = 6C. a = 1, b = 9D. a = 2, b = 8
easyAsked in NIMCET 2026View solution ›
Q6.
Let $A_{k}$ be the arithmetic mean of the squares of the first $k$ natural numbers. If $\sum_{k = 1}^{n} (6 A_{k} - 3 k) = 31$ , find the value of $n$ .
A. 3B. 2C. 4D. 1
mediumAsked in NIMCET 2026View solution ›
Q7.
Consider the sequence 72, 69, 66, … The numbers continue in the same pattern as long as they remain positive. What will be the maximum possible sum of the terms of this sequence?
A. 900B. 897C. 882D. 903
easyAsked in NIMCET 2026View solution ›
Q8.
The sum of $n$ terms of the series $1^{2} + (1^{2} + 2^{2}) + (1^{2} + 2^{2} + 3^{2}) + \dots$ is
A. $\frac{n ( n + 1 ) ( n + 2 )}{12}$ B. $\frac{n ( n + 1 ) ( n + 2 ) ^{2}}{12}$ C. $\frac{n ^{2} ( n + 1 ) ( n + 2 )}{12}$ D. $\frac{n ( n + 1 ) ^{2} ( n + 2 )}{12}$
hardAsked in AMU MCA 2016View solution ›
Q9.
If the angles of a triangle are in Arithmetic Progression, then the measures of one of the angles in radians is
A. $\frac{π}{6}$ B. $\frac{π}{3}$ C. $\frac{π}{2}$ D. $\frac{2 π}{3}$
easyAsked in MAH MCA CET 2024View solution ›
Q10.
The sum of infinitely many terms of a decreasing GP is equal to the greatest value of the function $f (x) = x^{3} + 3 x - 9$ in the interval $[- 2, 3]$ , and the difference between the first two terms is $f' (0)$ . Then the common ratio of the GP is
A. $- \frac{2}{3}$ B. $\frac{4}{3}$ C. $\frac{2}{3}$ D. $- \frac{4}{3}$
mediumAsked in NIMCET 2023 +1View solution ›
Q11.
If a, b, c, d are in HP and the arithmetic mean of ab, bc, cd is 9, then which of the following is the value of ad?
A. 3B. 9C. 12D. 4
mediumAsked in NIMCET 2023View solution ›
Q12.
In a G.P. consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of the G.P. is
A. $\frac{1 - 5}{2}$ B. $\frac{5}{2}$ C. $5$ D. $\frac{5 - 1}{2}$
mediumAsked in NIMCET 2013View solution ›
Q13.
The sum of n terms of an arithmetic series is 216. The value of the first term is n and the value of the $n^{t h}$ term is 2n. The common difference d is
A. 1B. $\frac{2}{3}$ C. $\frac{3}{2}$ D. $\frac{12}{11}$
mediumAsked in NIMCET 2013View solution ›
Q14.
The sum of integers between 200 and 400 that are multiples of 7 is
A. 8729B. 8700C. 8972D. 8279
easyAsked in NIMCET 2013View solution ›
Q15.
Sum of 20 terms of the series $- 1^{2} + 2^{2} - 3^{2} + 4^{2} - \dots$ is
A. 180B. 200C. 210D. 220
easyAsked in NIMCET 2013View solution ›
Q16.
If $H$ is the harmonic mean between $P$ and $Q$ , then $\frac{H}{P} + \frac{H}{Q}$ is
A. 2B. $\frac{P + Q}{Q}$ C. $\frac{P Q}{P + Q}$ D. None of these
easyAsked in NIMCET 2012View solution ›
Q17.
If $a, b, c$ are positive, $c > a$ , and $a, b, c$ are in H.P., then $l o g (a + c) + l o g (a - 2 b + c)$ is equal to
A. $2 l o g (c - b)$ B. $2 l o g (a + c)$ C. $2 l o g (c - a)$ D. $l o g a + l o g b + l o g c$
mediumAsked in NIMCET 2011View solution ›
Q18.
The sum $11^{2} + 12^{2} + 13^{2} + \dots + 30^{2}$ is
A. $9070$ B. $\frac{2}{82}$ C. $1080$ D. $9700$
easyAsked in NIMCET 2011View solution ›
Q19.
If $a, b, c$ are in A.P., $p, q, r$ are in H.P. and $a p, b q, cr$ are in G.P., then $\frac{p}{r} + \frac{r}{p}$ is equal to
A. $\frac{a}{c} - \frac{c}{a}$ B. $\frac{a}{c} + \frac{c}{a}$ C. $\frac{b}{q} - \frac{a}{p}$ D. $\frac{b}{q} + \frac{a}{p}$
hardAsked in NIMCET 2010View solution ›
Q20.
Find $k$ in the equation $x^{3} - 6 x^{2} + k x + 64 = 0$ if its roots are in geometric progression.
A. $24$ B. $16$ C. $- 16$ D. $- 24$
mediumAsked in NIMCET 2009View solution ›

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