Question

Find the sum of the first 30 terms in the sequence in #2. (Sequence is 16, 7, -2, …) Just need sum of first 30 solved :)

Answer 1

The sequence is arithmetic, since the forward difference between consecutive terms is -9.

7 - 16 = -9

-2 - 7 = -9

etc.

This means the sequence has the formula

$a_n=16-9(n-1)=25-9n$

The sum of the first 30 terms is

$\displaystyle\sum_{n=1}^{30}a_n=25\sum_{n=1}^{30}1-9\sum_{n=1}^{30}n$

Recall the formulas,

$\displaystyle\sum_{k=1}^n1=\underbrace{1+1+\cdots+1}_{n\text{ times}}=n$

$\displaystyle\sum_{k=1}^nk=1+2+3+\cdots+n=\frac{n(n+1)}2$

Then the sum we want is

$\displaystyle\sum_{n=1}^{30}a_n=25\cdot30-\frac{9\cdot30\cdot31}2=\boxed{-3435}$