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Progression & Series - Past Year Questions

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Q1:

Given that 1+122+132+142+...=π261 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}, the value of 1+132+152+172+...1 + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + ... is
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Q2:

The sum of the first 5 terms of a geometric progression is the same as the sum of the first 7 terms of the same progression. If the sum of the first 9 terms is 24, then the 4th term of the progression is
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Q3:

Let S1={100,105,110,115,...} S_1 = \{100, 105, 110, 115, ... \} and S2={100,95,90,85,...} S_2 = \{100, 95, 90, 85, ... \} be two series in arithmetic progression. If ak a_k and bk b_k are the k k -th terms of S1 S_1 and S2 S_2 , respectively, then ∑k=120akbk \sum_{k=1}^{20} a_k b_k equals __________.
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Q4:

If the sum of the first 2121 terms of the sequence: ln⁡ab,ln⁡abb,ln⁡ab2,ln⁡ab2b,…\ln \frac{a}{b}, \ln \frac{a}{b \sqrt{b}}, \ln \frac{a}{b^{2}}, \ln \frac{a}{b^{2} \sqrt{b}}, \ldots is ln⁡ambn\ln \frac{a^{m}}{b^{n}}, then the value of m+nm+n is \qquad
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Q5:

The terms of a geometric progression are real and positive. If the pp-th term of the progression is qq and the qq-th term is pp, then the logarithm of the first term is
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Q6:

The sum of a given infinite geometric progression is 80 and the sum of its first two terms is 35. Then the value of nn for which the sum of its first nn terms is closest to 100, is
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Q7:

Let a1,a2,a3a_{1}, a_{2}, a_{3} be three distinct real numbers in geometric progression. If the equations a1x2+2a2x+a3=0a_{1} x ^ 2 + 2a_{2}x + a_{3} = 0 and b1x2+2b2x+b3=0b_{1} x ^ 2 + 2b_{2}x + b_{3} = 0 have a common root, then which of the following is necessarily true?
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Q8:

A person standing at the centre of an open ground first walks 32 meters towards the east, takes a right turn and walks 16 meters, takes another right turn and walks 8 meters, and so on. How far will the person be from the original starting point after an infinite number of such walks in this pattern?
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Q9:

If f(n)=1+2+3+⋯+(n+1)f(n)= 1 + 2 + 3 +\cdots+(n+1) and g(n)=∑k=1k=n1f(k)g(n)= \sum_{k=1}^{k=n} \dfrac{1}{f(k)}, then the least value of nn for which g(n)g(n) exceeds the value 99100\dfrac{99}{100} is:
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Q10:

The sum of the first 15 terms in an arithmetic progression is 200, while the sum of the next 15 terms is 350. Then the common difference is
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Q11:

The numbers −16,2x+3−22x−1−16,22x−1+16-16,2^{x+3}-2^{2 x-1}-16,2^{2 x-1}+16 are in an arithmetic progression. Then xx equals ________.
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Q12:

The 3rd ,14th 3^{\text {rd }}, 14^{\text {th }} and 69th 69^{\text {th }} terms of an arithmetic progression form three distinct and consecutive terms of a geometric progression. If the next term of the geometric progression is the nth n^{\text {th }} term of the arithmetic progression, then nn equals ________.
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Q13:

A new sequence is obtained from the sequence of positive integers (1,2,3,…)(1,2,3, \ldots) by deleting all the perfect squares. Then the 2022nd 2022^{\text {nd }} term of the new sequence is ________.
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Q14:

Let SnS_n be sum of the first nn terms of an A.P. If S5=S9S_5 = S_9, what is the ratio of a3:a5a_3 : a_5
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Q15:

The sum up to 1010 terms of the series 1⋅3+5⋅7+9⋅11+...1 \cdot 3 + 5 \cdot 7 + 9 \cdot 11 + ... is
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Q16:

It is given that the sequence {xnx_n} satisfies x1=0,xn+1=xn+1+2√(1+xn)x_1 = 0, x_{n+1} = x_n + 1 + 2√(1+x_n) for n=1,2,...n = 1,2,... Then x31x_{31} is _______
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Q17:

If 112+122+132+…\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \ldots up to ∞=π26\infty = \frac{\pi^2}{6}, then the value of 112+132+152+…\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \ldots up to ∞\infty is
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Q18:

If (1+x−2x2)6=A0+∑r=112Arxr(1 + x - 2x^2)^6 = A_0 + \sum_{r=1}^{12} A_r x^r, then the value of A2+A4+A6+⋯+A12A_2 + A_4 + A_6 + \cdots + A_{12} is
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Q19:

The number of terms common to both the arithmetic progressions 2,5,8,11,...,1792, 5, 8, 11, ..., 179 and 3,5,7,9,...,1013, 5, 7, 9, ..., 101 is
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Q20:

Let α,β\alpha, \beta be the roots of x2−x+p=0x^2 - x + p = 0 and γ,δ\gamma, \delta be the roots of x2−4x+q=0x^2 - 4x + q = 0 where p and q are integers. If α,β,γ,δ\alpha, \beta, \gamma, \delta are in geometric progression then p+qp + q is
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Q21:

There are numbers a1,a2,a3,…,ana_1, a_2, a_3, \ldots, a_n each of them being +1+1 or −1-1. If it is known that a1a2+a2a3+a3a4+…an−1an+ana1=0a_1 a_2 + a_2 a_3 + a_3 a_4 + \ldots a_{n-1} a_n + a_n a_1 = 0 then
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Q22:

Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The 6389th digit in this sequence is
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Q23:

Given below are two statements, one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) :
The sum of nn terms of the Progression
1+12+122+123+1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+ is 2n−1−12n−1\frac{2^{n-1}-1}{2^{n-1}}.
Reason (R) :
Sum of a geometric series having nn terms is given by Sn=a(1−rn)1−rS_{n}=\frac{a\left(1-r^{n}\right)}{1-r}, where aa is the 1st 1^{\text {st }} term and rr is the common ratio.
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Q24:

Assertion [A]: Sum of the first hundred even natural numbers divisible by 5 is 45050.
Reason (R): Sum of the first n-terms of an Arithmetic Progression is given by S=(n/2)∗(a+l)S = (n/2) *(a + l) where a=first term, l=last term.
Choose the correct answer from the options given below.
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Q25:

The sum of n−n- terms of sequence 11×2+12×3+13×4……\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4} \ldots \ldots. Is
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Q26:

If the mthm^{\text{th}} term of an arithmetic progression is 1n\frac{1}{n} and the nthn^{\text{th}} term is 1m\frac{1}{m}, then the mnthmn^{\text{th}} term of this progression will be
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Q27:

The value of (19−8−18−7+17−6−16−5+15−4)(\frac{1}{\sqrt{9}-\sqrt{8}} - \frac{1}{\sqrt{8}-\sqrt{7}} + \frac{1}{\sqrt{7}-\sqrt{6}} - \frac{1}{\sqrt{6}-\sqrt{5}} + \frac{1}{\sqrt{5}-\sqrt{4}}) is
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Q28:

The height of nineteen people of comic book is in Arithmetic progression. The average height of them is 19 feet. If the tallest is 37 feet. Then what is the height of the shortest?
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Q29:

The sum of third and ninth term of an A.P is 8. Find the sum of the first 11 terms of the progression.
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Q30:

Given (A=265)(A = 2^{65}) and (B=264+263+262+...+20)(B = 2^{64} + 2^{63} + 2^{62} + ... + 2^0), which of the following is true?
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Q31:

If log⁡2,log⁡(2x−1),log⁡(2x+3)\log 2, \log (2x - 1), \log (2x + 3) are in A.P, then xx is equal to ____
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Progression & Series - Past Year Questions (Free PDF Download)

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