Question

If S₁ = a₂ + a₄ + a₆ + ... + upto 100 terms and S₂ = a₁ + a₃ + a₅ + ... + upto 100 terms of a certain A.P., then its common difference is 

(a) S₁ – S₂ 

(b) S₂ – S₁ 

(c) \(\frac{S_1 - S_2}{2}\)

(d) None of these

Answer 1

Answer : (d) None of these

 Given that, 

S₁ = a₂ + a₄ + a₆ + ..... + upto 100 terms …(i) 

S₂ = a₁ + a₃ + a₅ + ..... + upto 100 terms …(ii) 

Let d be the common difference of the given A.P. Then, 

d = a₂ – a₁ = a₃ – a₂ = a₄ – a₃ = …(iii)

Subtracting eqn (ii) from eqn (i), we have

S₁ – S₂ = (a₁ – a₂) + (a₄ – a₃) + (a₆ – a₅) +.... + upto 100 terms 

= d + d + d + .... + upto 100 terms = 100 d

⇒ d = \(\frac{S_1 - S_2}{100}\)