004 Sample Final A, Problem 7

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

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

Solution:

Step 1:
The first term in the arithmetic sequence is $A_{1}=10$ and the common difference is $d=-3$.

Step 2:
The 20th term of this sequence is $A_{20}=10+-3(20-1)=10-3(19)=10-57=-47$.

Step 3:
The sum of the first twenty terms is $S_{20}={\frac {20}{2}}(10+-47)=10(-37)=-370$.

Final Answer:
$-370$

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