Question

Find the sum 38+39+40+41...+114+115

Answer 1

It seems like you want to find the sum of 38 to 115:

$\displaystyle \large{38 + 39 + 40 + 41 + ... + 114 + 115}$

If we notice, this is arithmetic series or the sum of arithmetic sequences.

To find the sum of the sequences, there are three types of formulas but I will demonstrate only one and the best for this problem.

$\displaystyle \large{S_n = \frac{n(a_1+a_n) }{2} }$

This formula only applies to the sequences that have the common difference = 1.

Given that a1 = first term of sequence/series, n = number of terms and a_n = last term

We know the first term which is 38 and the last term is 115. The problem here is the number of sequences.

To find the n, you can use the following formula.

$\displaystyle \large{n = (a_n - a_1) + 1}$

Substitute an = 115 and a1 = 38 in the formula of finding n.

$\displaystyle \large{n = (115 - 38) + 1} \\ \displaystyle \large{n = (77) + 1} \\ \displaystyle \large{n = 78}$

Therefore the number of sequences is 78.

Then we substitute an = 115, a1 = 38 and n = 78 in the sum formula.

$\displaystyle \large{S_{78} = \frac{78(38+115) }{2} } \\ \displaystyle \large{S_{78} = \frac{39(38+115) }{1} } \\ \displaystyle \large{S_{78} = 39(153) } \\ \displaystyle \large \boxed{S_{78} = 5967}$

Hence, the sum is 5967.