Question

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

Answer 1

Decompose the terms in the sum as

$4 + 15 + 26 + \cdots + 213 = 4 + (4 + 11) + (4 + 2\times11) + \cdots + (4 + 19\times11)$

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

$4 + 15 + 26 + \cdots + 213 = 4\times20 + (0+1+2+\cdots+19)\times11$

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

$2S = (1 + 19) + (2 + 18) + \cdots + (18 + 2) + (19 + 1) = 19\times20 \implies S=19\times10=190$

So, our sum evaluates to

$4 + 15 + 26 + \cdots + 213 = 4\times20 + 190\times11 = \boxed{2170}$