Sequence and series JEE Main 2025 PYQ with Solution

Question :

In an Arithmetic Progression , If S₄₀ = 1030 and S₁₂ = 57, then S₃₀ – S₁₀ is equal to

Options :

(a) 505

(b) 515

(c) 510

(d) 525

Sequence and series JEE Main 2025 PYQ with Solution

Answer :

Step : 1 Given,

S₄₀ = 1030

$\frac{40}{2}$(2a + 39d ) = 1030

40a + 780d = 1030

4a + 78d = 103 ——————————1

S₁₂ = 57

$\frac{12}{2}$(2a + 11d ) = 57

6( 2a + 11d ) = 57

4a + 22d = 19————————————-2

Step : 2

Solving 1 and 2 equation

4a + 78d – 4a – 22d = 103 – 19

56d = 84

d = $\frac{84}{56}$

d = $\frac{3}{2}$

Step : 3

substitute d = $\frac{3}{2}$ in 2nd equation

4a + 33 = 19

4a = 19 – 33

a = –$\frac{14}{4}$

a = –$\frac{7}{2}$

Step : 4

S₃₀ – S₁₀ = $\frac{30}{2}$( 2a + 29d ) – $\frac{10}{2}$( 2a + 9d )

= 30a + 435d – 10a – 45d

= 20a + 390d

Step : 5

substitute a = –$\frac{7}{2}$ , d = $\frac{3}{2}$

= 20( –$\frac{7}{2}$ ) + 390 ( $\frac{3}{2}$ )

= -70 + 585 = 515

Correct answer : Option (b)

Practice Questions :

1. In a geometric series of positive terms the difference between the fifth and fourth terms is 576, and the difference between the second and first terms is 9 what is the sum of the first five terms of this series ?

Options :

(A) 1061

(B) 1023

(C) 1024

(D) 768

(E) None of these

Answer :

Given

ar⁵ – ar⁴ = 576

ar⁴ ( r – 1 ) = 576 ——————– 1

ar² – a = 9

a ( r – 1) = 9 ——————— 2

Divide 1 / 2

$\frac{ar^4 [ r – 1] }{ar [ r -1]}$ = $\frac{576}{9}$

r³ = 64

r = 4

substitude r = 4 in 2nd equation

a (3) = 9

3a = 9

a = 3

S_n = $\frac{ a (r^n – 1)}{ r -1}$

S₅ = $\frac{ 3 ( 4^5 – 1)}{3}$

S₅ = $\frac{ 3 [ 1024 – 1]}{3}$

S₅ = 1023

Correct answer : Option (B)