Question

Write a program to display the series upto n terms:25,100,225............
no copy or spam is allowed!!

Answer 1

Answer:

Refer to the attachment

Answer 2

Answer:

Let

$s = 25 + 100 + 225 + .... \\ s = 25(1 + 4 + 9.....)$

So we need some of series

1 , 4 , 9 , 16.....up to n terms

Common differences are in Arithmetic series , in other words it is an Arithmetic geometric series.

so general term of this kind of series is given by

$a_n = a {n}^{2} + bn + c$

Now n = 1 , a_1 = 1

n = 2 , a_2 = 4

n= 3 , a_3 = 9

$1 = a + b + c \\ 4 = 4 a + 2b + c \\ 9 = 9a + 3b + c$

solution is clear

$a = 1 \: \: \: \: \: b = 0 \: \: \: \: \: c = 0$

So general term of series is given by

$a_n = {n}^{2}$

So in our original series

$s = \sum _{k = 1} ^{n} 25 {k}^{2} \\ \\ s = 25 \sum _{k = 1} ^{n} {k}^{2} \\ \\ s = 25( \frac{n(n + 1)(2n + 1)}{6} )$