Question

the sums of first n terms of three AP are s1 , s2 and s3 . the first term of each is 5 and their common difference are 2,4, and 6 respectively. prove that s1 +s3 =2S2.

Answer 1

this is the answer hope it is correct

Answer 2

According to the Question

$\bf\huge S_{1} = \frac {n}{2} [2 + (n - 1)1]$

$\bf\huge S_{2} = \frac{n}{2}[2 + (n - 1)2]$

$\bf\huge S_{3} = \frac{n}{2}[2 + (n - 1)3]$

Now

$\bf\huge S_{1} + S_{3} = \frac{n}{2}[4 + (n - 1)4]$

$\bf\huge S_{1} + S_{3} = 2\times \frac{n}{2}[2 + (n - 1)2]$

$\bf\huge S_{1} + S_{3} = 2\times S_{2}$