Question

Find the sum of the first n terms of the series: 3 + 7 + 13 + 21 + 31 + …

Answer 1

The given series is 3 + 7 + 13 + 21 + 31 + …

S = 3 + 7 + 13 + 21 + 31 + …+ a_n–1 + a_n

S = 3 + 7 + 13 + 21 + …. + a_{n – 2} + a_{n – 1} + a_n

On subtracting both the equations, we obtain

S – S = [3 + (7 + 13 + 21 + 31 + …+ a_n–1 + a_n)] – [(3 + 7 + 13 + 21 +31 + …+ a_n–1)+ an]

S – S = 3 + [(7 – 3) + (13 – 7) + (21 – 13) + … + (a_n – a_n–1)] – a_n

0 = 3 + [4 + 6 + 8 + … (n –1) terms] – a_n  

a_n = 3 + [4 + 6 + 8 + … (n –1) terms]