Sequences and Series

21 solved problems from JEE previous year papers.

Arithmetic Progressions

Each Section of Each Class Plants as Many Trees as the Class in Which They Study — a Section of Class I Plants 1 Tree, Class II Plants 2 Trees, and So On Till Class XII. There Are Three Sections of Each Class. How Many Trees Will Be Planted

NCERT Class 10 Arithmetic Progressions problem: each of the 3 sections of Classes I to XII plants trees equal to its class…
Arithmetic Progressions

A Spiral Is Made Up of Successive Semicircles With Centres Alternately at A and B, of Radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, … What Is the Total Length of Such a Spiral Made Up of Thirteen Consecutive Semicircles

NCERT Class 10 Arithmetic Progressions problem: a spiral of 13 successive semicircles with radii increasing by 0.5cm each time.…
Sequences and Series

If S₂₀ = 30 and S₃₀ = 20 for an Arithmetic Progression, Find S₅₀

Given the sum of the first 20 terms of an AP is 30 and the sum of the first 30 terms is 20, find S50 by solving for the first…
Sequences and Series

m Arithmetic Means Are Inserted Between 1 and 31 So That the Ratio of the 7th and (m−1)th Means Is 5:9. Find the Value of m

m arithmetic means are inserted between 1 and 31 so the ratio of the 7th and (m−1)th means is 5:9. Find m by expressing both…
Sequences and Series

Let a, ar, ar², … Be an Infinite G.P. If the Sum of arⁿ From n = 0 to ∞ Is 57 and the Sum of a³r³ⁿ Is 9747, Then a + 18r Is Equal To

For an infinite GP a, ar, ar², … with Σar^n=57 and Σa³r^3n=9747, find a+18r by dividing the cubed-series sum by the cube of…
Sequences and Series

The Greatest Positive Integer k for Which 49ᵏ + 1 Is a Factor of the Sum 49¹²⁵ + 49¹²⁴ + ⋯ + 49² + 49 + 1 Is

Find the greatest positive integer k such that 49^k+1 divides 1+49+49²+…+49¹²⁵, by writing the sum as a GP and factoring it…
Sequences and Series

In an AP, if m·tₘ = n·tₙ With m ≠ n, Find tₘ₊ₙ

In an arithmetic progression, if m times the mth term equals n times the nth term (m≠n), prove that the (m+n)th term must equal…
Sequences and Series

Find the Positive Integer n for Which 2·2² + 3·2³ + 4·2⁴ + ⋯ + n·2ⁿ = 2ⁿ⁺¹⁰

Find the positive integer n satisfying 2·2²+3·2³+…+n·2ⁿ=2^(n+10), by summing the series via the standard shift-and-subtract…
Sequences and Series

If Three Positive Real Numbers a, b, c Are in AP Such That abc = 4, Then the Minimum Value of b Is

Given positive reals a,b,c in AP with abc=4, find the minimum value of b using the AM-GM inequality on a, b, c.
Sequences and Series

The Sum of the Series 1/(1·2) − 1/(2·3) + 1/(3·4) − ⋯ up to Infinity Is

Find the sum of the alternating series 1/(1·2)−1/(2·3)+1/(3·4)−… to infinity, by splitting each term via partial fractions…
Sequences and Series

Two APs: s1=t1≠0, s2=2t2 — Find (s2−s1)/(t2−t1)

Two arithmetic sequences share a first term and satisfy s2=2t2. Given the sum of 10 terms of one equals the sum of 15 terms of…
Sequences and Series

If Sm=Sn for an AP, Prove Sm+n=0

Prove that if the sum of the first m terms of an AP equals the sum of the first n terms (m≠n), then the sum of the first m+n…
Sequences and Series

If the First Term of an Arithmetic Progression Is 3 and the Sum of the First 4 Terms Is Equal to 1/5th of the Sum of the Next 4 Terms, Then the Sum of the First 20 Terms Is

An AP has first term 3, and the sum of its first 4 terms equals 1/5 of the sum of the next 4 terms. Find the sum of the first 20…
Sequences and Series

Limit of Σ(k³+6k²+11k+5)/(k+3)! From k=1 to n as n→∞

Evaluate the infinite series limit of (k³+6k²+11k+5)/(k+3)! by rewriting the numerator in terms of j=k+3 to reveal a telescoping…
Sequences and Series

AP: a1+(a5+a10+…+a2020)+a2024=2233, Find Sum to 2024 Terms

For an AP a1,…,a2024 with a1+(a5+a10+…+a2020)+a2024=2233, find a1+a2+…+a2024 by summing the selected terms as…
Sequences and Series

Sum of 1 + 6 + 9(1²+2²+3²)/7 + 12(1²+2²+3²+4²)/9 + … Up to 15 Terms

The sum of the series 1+6+9(1²+2²+3²)/7+12(1²+2²+3²+4²)/9+15(1²+…+5²)/11+… up to 15 terms is: (1)7820 (2)7520 (3)7830…
Sequences and Series

G.P. of Increasing Positive Terms, a1a5=28 and a2+a4=29, Find a6

Let a1,a2,a3,… be a G.P. of increasing positive terms. If a1a5=28 and a2+a4=29, then a6 is equal to: (A) 628 (B) 812 (C) 526 (D)…
Sequences and Series

The Sum to 10 Terms of the Series 1/(1+1²+1⁴) + 2/(1+2²+2⁴) + 3/(1+3²+3⁴) + … Is

The sum to 10 terms of the series 1/(1+1²+1⁴) + 2/(1+2²+2⁴) + 3/(1+3²+3⁴) + … is (A) 56/111 (B) 58/111 (C) 55/111 (D)…
Sequences and Series

How to Find the Sum of 7 + 77 + 777 + … Up to n Terms

A step-by-step derivation of the sum of the series 7 + 77 + 777 + … up to n terms, using the repunit-pattern trick to write…
Sequences and Series

If Sn=1/2+1/6+1/12+… to n Terms, A.P. Sum is √(2026·S2025)

A JEE Main 2025 problem combining a telescoping series with an arithmetic progression, solved by simplifying Sn first, then using…
Arithmetic Progressions

3-Digit Numbers Divisible by 2 and 3, Not 4 and 9

How many 3-digit numbers are divisible by both 2 and 3, but not by 4 or 9? Solved using LCMs and arithmetic progressions, JEE…