Question

The sum of series of 4+ 15 + 26 + ….. + 213

Answer 1

Decompose the terms in the sum as

`4 + 15 + 26 + \cdots + 213 = 4 + (4 + 11) + (4 + 2*11) + \cdots + (4 + 19*11)`

so there are 20 terms in the sum. Then our sum simplifies to

`4 + 15 + 26 + \cdots + 213 = 4*20 + (0+1+2+\cdots+19)*11`

Now, if

`S = 1 + 2 + 3 + \cdots + 17 + 18 + 19`

we can reverse the order of terms to get the same sum,

`S = 19 + 18 + 17 + \cdots + 3 + 2 + 1`

so that doubling up the sum gives

`2S = (1 + 19) + (2 + 18) + \cdots + (18 + 2) + (19 + 1) = 19*20 \implies S=19*10=190`

So, our sum evaluates to

`4 + 15 + 26 + \cdots + 213 = 4*20 + 190*11 = \boxed{2170}`