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

Q21.
If $A_{1} = 3$ , $A_{2} = 5, 7, 9$ , $A_{3} = 11, 13, 15, 17, 19$ , $A_{4} = 21, 23, 25, 27, 29, 31, 33$ and so on, what is the average of the numbers of the set $A_{20}$ ?
A. 761B. 763C. 765D. 767
mediumAsked in NIMCET 2009View solution ›
Q22.
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 ›
Q23.
From a height of $16$ metres a ball fell down and each time it bounces half the distance back. What is the total distance travelled?
A. $45$ mB. $\infty$ C. $48$ mD. $24$ m
easyAsked in NIMCET 2008View solution ›
Q24.
An arithmetic progression has 3 as its first term. Also, the sum of the first 8 terms is twice the sum of the first 5 terms. Then what is the common difference?
A. $\frac{3}{4}$ B. $\frac{1}{2}$ C. $\frac{1}{4}$ D. $\frac{4}{3}$
easyAsked in NIMCET 2020View solution ›
Q25.
The four geometric means between 2 and 64 are
A. $\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}$ B. $4, 8, 16, 32$ C. $42, 8, 162, 32$ D. None of the above
easyAsked in NIMCET 2021View solution ›
Q26.
Which term of the series $\frac{5}{3}, \frac{5}{4}, \frac{1}{5}, \dots$ is $\frac{5}{13}$ ?
A. 12B. 11C. 10D. 9
easyAsked in NIMCET 2022View solution ›
Q27.
In a harmonic progression, the $p^{t h}$ term is $q$ and the $q^{t h}$ term is $p$ . Then the $(pq)^{t h}$ term is
A. 0B. $pq$ C. 1D. $pq (p + q)$
mediumAsked in NIMCET 2022View solution ›
Q28.
If $a$ , $b$ , $c$ are in geometric progression, then $l o g_{a x} a$ , $l o g_{b x} a$ and $l o g_{c x} a$ are in
A. Arithmetic progressionB. Geometric progressionC. Harmonic progressionD. Arithmetico-geometric progression
mediumAsked in NIMCET 2015View solution ›
Q29.
The value of the sum $\frac{1}{2 1 + 1 2} + \frac{1}{3 2 + 2 3} + \frac{1}{4 3 + 3 4} + \dots + \frac{1}{25 24 + 24 25}$ is
A. $\frac{9}{10}$ B. $\frac{4}{5}$ C. $\frac{14}{15}$ D. $\frac{7}{15}$
mediumAsked in NIMCET 2015View solution ›
Q30.
The harmonic mean of two numbers is 4. Their arithmetic mean A and geometric mean G satisfy the relation $2 A + G^{2} = 27$ . Then the two numbers are
A. 4 and 2B. 6 and 3C. 5 and 7D. 4 and 1
mediumAsked in NIMCET 2017View solution ›
Q31.
Three positive numbers whose sum is 21 are in arithmetic progression. If 2, 2, 14 are added to them respectively then the resulting numbers are in geometric progression. Then which of the following is not among the three numbers?
A. 25B. 13C. 1D. 7
mediumAsked in NIMCET 2017View solution ›
Q32.
The sum to infinity of a geometric progression is twice the sum of its first two terms. What are the possible values of the common ratio?
A. $\pm \frac{1}{2}$ B. $\pm \frac{1}{2}$ C. $\pm \frac{1}{3}$ D. $\pm \frac{1}{3}$
easyAsked in NIMCET 2018View solution ›
Q33.
Suppose that m and n are fixed numbers such that the mth term of an HP is equal to n and the nth term is equal to m $(m \neq = n)$ . Then the $(m + n)$ th term is
A. $\frac{m + n}{mn}$ B. $\frac{mn}{m + n}$ C. $\frac{m + n}{n}$ D. $\frac{m + n}{m}$
mediumAsked in NIMCET 2018View solution ›
Q34.
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 ›
Q35.
If $x$ , $2 x + 2$ , $3 x + 3$ are the first three terms of a geometric progression, then the 4th term of the geometric progression is:
A. -13.5B. 13.5C. -27D. 27
easyAsked in NIMCET 2019View solution ›
Q36.
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 ›
Q37.
If $x = 1 + 62 + 64 + 68 + 616 + 632$ , then $(1 + \frac{1}{x})^{24} =$
A. 1B. 4C. 16D. 24
mediumAsked in NIMCET 2024View solution ›
Q38.
The number of solutions of $5^{1 + ∣ s in x ∣ + ∣ s in x ∣^{2} + \dots} = 25$ for $x \in (- π, π)$ is
A. 2B. 0C. 4D. $\infty$
mediumAsked in NIMCET 2024View solution ›
Q39.
If one AM (arithmetic mean) $a$ and two GMs (geometric means) $p$ and $q$ are inserted between any two positive numbers, then the value of $p^{3} + q^{3}$ is
A. $2 a pq$ B. $\frac{pq}{a}$ C. $\frac{2 pq}{a}$ D. $p + q + a$
mediumAsked in NIMCET 2024View solution ›
Q40.
The value of the series $\frac{2}{3 !} + \frac{4}{5 !} + \frac{6}{7 !} + \dots$ is
A. $2 e^{- 2}$ B. $e^{- 2}$ C. $e^{- 1}$ D. $2 e^{- 1}$
mediumAsked in NIMCET 2024View solution ›

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