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.
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 ›
Q3.
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 ›
Q4.
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 ›
Q5.
If $H_{1}, H_{2}, \dots, H_{n}$ are $n$ harmonic means between $a$ and $b$ $(b \neq = a)$ , then $\frac{H _{1} + a}{H _{1} - a} + \frac{H _{n} + b}{H _{n} - b}$ is equal to
A. $2 n$ B. $n + 1$ C. $n - 1$ D. $2 n + 1$
hardAsked in NIMCET 2008 +1View solution ›
Q6.
If a, b, c are in GP and $l o g a - l o g 2 b$ , $l o g 2 b - l o g 3 c$ and $l o g 3 c - l o g a$ are in AP, then a, b, c are the lengths of the sides of a triangle which is
A. Acute angledB. Obtuse angledC. Right angledD. Equilateral
hardAsked in NIMCET 2019View solution ›
Q7.
If $a, a_{1}, a_{2}, \dots, a_{2 n - 1}, b$ are in AP, $a, b_{1}, b_{2}, \dots, b_{2 n - 1}, b$ are in GP and $a, c_{1}, c_{2}, \dots, c_{2 n - 1}, b$ are in HP, where a, b are positive, then the equation $a_{n} x^{2} - b_{n} x + c_{n} = 0$ has its roots
A. Real and equalB. Real and unequalC. ImaginaryD. One real and one imaginary
hardAsked in NIMCET 2019View solution ›
Q8.
Suppose $t_{1}, t_{2}, t_{3}, \dots$ are in AP such that $\sum_{l = 0}^{18} t_{3 l + 1} = 1197$ and $t_{7} + 3 t_{22} = 174$ . If $\sum_{l = 1}^{9} t_{l}^{2} = 947 b$ , then the value of $b$ is
A. 1B. 2C. 3D. 5
hardAsked in NIMCET 2025View solution ›