Question

if unity is added to the sum of any number of terms of the A.P. 3,5,7,9,..... the resulting sum is?​

Answer 1

$\huge\sf\blue{\underline{\underline{\purple{Solution}}}}$

Let 1 add as a first term of the Arithmetic progression series 3, 5, 7, 9, .... and n be the total numbers then the series becomes,

1, 3, 7, 9, ...., (2n - 1)

Here, first term a = 1, difference d = 3 - 2 = 2

So, the sum of the series,

= n/2[2a + (n - 1)d]

= n/2[2 × 1 + (n - 1)2]

= n/2[2 + 2n - 2]

= n/2[ 2n ]

= n^2

So, the sum of the series is a perfect square.