Question

If S₁ be the sum of (2n + 1) terms of an A.P. and S₂ sum of its odd terms, then prove that: S₁: S₂ = (2n + 1) : (n + 1).

Answer 1

To prove : S₁: S₂ = (2n + 1) : (n + 1)

We know that the sum of AP is given by the formula :

s = \(\frac{n}{2}\)(2a + (n-1)d)

Substituting the values in the above equation,

s₁ = \(\frac{2n+1}{2}\)(2a+2nd)

For the sum of odd terms, it is given by,

s₂ = a₁ + a₃ + a₅ + …. a_{2n + 1} 

s₂ = a + a + 2d + a + 4d + … + a + 2nd 

s₂ = (n + 1)a + n(n + 1)d 

s₂ = (n + 1) (a + nd) 

Hence,

s₁ : s₂ = \(\frac{2n+1}{n+1}\)