# 004 Sample Final A, Problem 7

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Given a sequence ${\displaystyle 10,7,4,1,\ldots }$ use formulae on the back page to compute ${\displaystyle S_{20}}$

Foundations
1) What is the formula for the nth term of an arithmetic sequence?
2) What is the formula for the sum of the first n terms of an arithmetic sequence?
1) The nth term of an arithmetic sequence is ${\displaystyle A_{n}=A_{1}+d(n-1)}$ where ${\displaystyle A_{1}}$ is the first term of the sequence and ${\displaystyle d}$ is the common difference.
2) The sum of the first n terms of an arithmetic sequence is ${\displaystyle S_{n}={\frac {n}{2}}(A_{1}+A_{n})}$.

Solution:

Step 1:
The first term in the arithmetic sequence is ${\displaystyle A_{1}=10}$ and the common difference is ${\displaystyle d=-3}$.
Step 2:
The 20th term of this sequence is ${\displaystyle A_{20}=10+-3(20-1)=10-3(19)=10-57=-47}$.
Step 3:
The sum of the first twenty terms is ${\displaystyle S_{20}={\frac {20}{2}}(10+-47)=10(-37)=-370}$.
${\displaystyle -370}$