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Sequences and Series — JEE Main & Advanced IOQM PYQs

38 previous year questions from Sequences and Series with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

38 questionsIOQMSolutions on every page
1

A finite set M of positive integers consists of distinct perfect squares and the number 92 . The average of the numbers in M is 85 . If we remove 92 from M , the average drops to 8

Olympiad workbook Solution View → 2

If _ k=1 ^N 2 k+1 (k^2+k )^2 =0.9999 then determine the value of N .

Olympiad workbook Solution View → 3

A group of women working together at the same rate can build a wall in 45 hours. When the work started, all the women did not start working together. They joined the work over a pe

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Let m be the smallest positive integer such that m^2+(m+1)^2+ +(m+10)^2 is the square of a positive integer n . Find m+n .

Olympiad workbook Solution View → 5

Let u , v , w be real numbers in geometric progression such that u > v > w . Suppose u ⁴⁰= v ^ n = w ⁶⁰ . Find the value of n .

Olympiad workbook Solution View → 6

The sequence a_n _ n 0 is defined by a₀=1, a₁=-4 and a_ n+2 =-4 a_ n+1 -7 a_n , for n 0 . Find the number of positive integer divisors of a₅₀^2-a₄₉ a₅₁ .

Olympiad workbook Solution View → 7

Let the sum _ n=1 ^9 1 n(n+1)(n+2) written in its lowest terms be p q . Find the value of q-p .

Olympiad workbook Solution View → 8

Consider the set T of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let T be the triangle with the least perimeter. If a ^ is the l

Olympiad workbook Solution View → 9

Starting with a positive integer M written on the board, Alice plays the following game: in each move, if x is the number on the board, she replaces it with 3 x+2 . Similarly, star

Olympiad workbook Solution View → 10

_ k=1 ⁴⁰ ( 1+ 1 k^2 + 1 (k+1)^2 )=a+ b c where a, b, c N , b < c, gcd (b, c)=1 , then what is the value of a+b ?

Olympiad workbook Solution View → 11

Integers 1,2,3 . n where n >2 , are written on a board. Two numbers m , k such that 1 < m < n .1 < k < n are removed and the average of the remaining numbers is found to be 17 . Wh

Olympiad workbook Solution View → 12

Five distinct 2-digit numbers are in a geometric progression. Find the middle term.

Olympiad workbook Solution View → 13

Suppose x is a positive real number such that x .[ x ] and x are in the geometric progression. Find the least positive integer n such that x^n>100 . (Here [x] denotes the integer p

Olympiad workbook Solution View → 14

Each of the numbers x ₁, x ₂, . x ₁₀₁ is 1 . What is the smallest positive value of _ 1 i j 101 x_i x_j ?

Olympiad workbook Solution View → 15

A natural number k >1 is called good if there exist natural numbers a₁ < a₂ < < a_k such that 1 a₁ + 1 a₂ + + 1 a_k =1 Let f(n) be the sum of the first n good numbers, n 1 . Find t

Olympiad workbook Solution View → 16

A natural number n is said to be good if n is the sum of r consecutive positive integers, for some r 2 . Find the number of good numbers in the set 1,2, , 100) .

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Let a₁=24 and form the sequence a_n, n 2 by a_n=100 a_ n-1 +134 . The first few terms are 24,2534,253534,25353534, What is the least value of n for which a _ n is divisible by 99 ?

Olympiad workbook Solution View → 18

What is the largest positive integer n such that a^2 b 29 + c 31 + b^2 c 29 + a 31 + c^2 a 29 + b 31 n(a+b+c) holds for all positive real numbers a,b,c.

Olympiad workbook Solution View → 19

What is the value of _ 1 i < j 10 i+j= odd (i+j)- _ 1 i < j 10 i+j= even (i+j) ?

Olympiad workbook Solution View → 20

_ k=1 ⁴⁰ ( 1+ 1 k^2 + 1 (k+1)^2 )=a+ b c where a, b, c N , b < c, gcd (b, c)=1 , then what is the value of a+b ?

Olympiad workbook Solution View → 21

A group of women working together at the same rate can build a wall in 45 hours. When the work started, all the women did not start working together. They joined the work over a pe

Olympiad workbook Solution View → 22

Consider the set T of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let T be the triangle with the least perimeter. If a ^ is the l

Olympiad workbook Solution View → 23

Let m be the smallest positive integer such that m^2+(m+1)^2+ +(m+10)^2 is the square of a positive integer n . Find m+n .

Olympiad workbook Solution View → 24

Starting with a positive integer M written on the board, Alice plays the following game: in each move, if x is the number on the board, she replaces it with 3 x+2 . Similarly, star

Olympiad workbook Solution View → 25

The sequence a_n _ n 0 is defined by a₀=1, a₁=-4 and a_ n+2 =-4 a_ n+1 -7 a_n , for n 0 . Find the number of positive integer divisors of a₅₀^2-a₄₉ a₅₁ .

Olympiad workbook Solution View → 26

If _ k=1 ^N 2 k+1 (k^2+k )^2 =0.9999 then determine the value of N .

Olympiad workbook Solution View → 27

Let the sum _ n=1 ^9 1 n(n+1)(n+2) written in its lowest terms be p q . Find the value of q-p .

Olympiad workbook Solution View → 28

Let u , v , w be real numbers in geometric progression such that u > v > w . Suppose u ⁴⁰= v ^ n = w ⁶⁰ . Find the value of n .

Olympiad workbook Solution View → 29

Five distinct 2-digit numbers are in a geometric progression. Find the middle term.

Olympiad workbook Solution View → 30

Integers 1,2,3 . n where n >2 , are written on a board. Two numbers m , k such that 1 < m < n .1 < k < n are removed and the average of the remaining numbers is found to be 17 . Wh

Olympiad workbook Solution View → 31

A finite set M of positive integers consists of distinct perfect squares and the number 92 . The average of the numbers in M is 85 . If we remove 92 from M , the average drops to 8

Olympiad workbook Solution View → 32

Suppose x is a positive real number such that x .[ x ] and x are in the geometric progression. Find the least positive integer n such that x^n>100 . (Here [x] denotes the integer p

Olympiad workbook Solution View → 33

Each of the numbers x ₁, x ₂, . x ₁₀₁ is 1 . What is the smallest positive value of _ 1 i j 101 x_i x_j ?

Olympiad workbook Solution View → 34

A natural number k >1 is called good if there exist natural numbers a₁ < a₂ < < a_k such that 1 a₁ + 1 a₂ + + 1 a_k =1 Let f(n) be the sum of the first n good numbers, n 1 . Find t

Olympiad workbook Solution View → 35

Let a₁=24 and form the sequence a_n, n 2 by a_n=100 a_ n-1 +134 . The first few terms are 24,2534,253534,25353534, What is the least value of n for which a _ n is divisible by 99 ?

Olympiad workbook Solution View → 36

A natural number n is said to be good if n is the sum of r consecutive positive integers, for some r 2 . Find the number of good numbers in the set 1,2, , 100) .

Olympiad workbook Solution View → 37

What is the largest positive integer n such that a^2 b 29 + c 31 + b^2 c 29 + a 31 + c^2 a 29 + b 31 n(a+b+c) holds for all positive real numbers a,b,c.

Olympiad workbook Solution View → 38

What is the value of _ 1 i < j 10 i+j= odd (i+j)- _ 1 i < j 10 i+j= even (i+j) ?

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