Let s=3 + 5 + 9 + 15 + 23 + …………. + n

By shifting each term by one

S = 3 + 5 + 9 + 15 + 23 + …………. + nth ………(1)

S = 3 + 5 + 9 + 15 + …………. + (n - 1)th + nth ……..(2)

by (1) - (2) we get

0 = 3 + 2 + 4 + 6 + 8 + …….nth - (n - 1)th - n

Nth = 3 + 2 + 4 + 6 + 8 + …….2(n - 1)th

Nth = 3 + 2(1 + 2 + 3 + 4 + …….(n - 1)th) ……….(3)

we know

Substituting the above-given value in (3)

nth = 3 + n² - n

general term = 3 + r² - r

thus

S = 3 + 5 + 9 + 15 + 23 + …………. + nth =

We know by property that:

∑axⁿ + bx^{n - 1} + cx^{n - 2}…….d₀=a∑xⁿ + b∑x^{n - 1} + c∑x^{n - 2}…….. + d₀∑1

(4)

We know

Thus substituting the above values in(4)