# 008A Sample Final A, Question 8

Question: Given a sequence ${\displaystyle 27,23,19,15,\ldots }$ use formulae to compute ${\displaystyle S_{10}}$ and ${\displaystyle A_{15}}$.

Foundations
1) Which of the ${\displaystyle S_{n}}$ formulas should you use?
2) What is the common ratio or difference?
3) How do you find the values you need to use the formula?
1) The variables in the formulae give a bit of a hint. The r stands for ratio, and ratios are associated to geometric series. This sequence is arithmetic, so we want the formula that does not involve r.
2) We determine the common difference by taking two adjacent terms in the sequence, say ${\displaystyle A_{1}}$ and ${\displaystyle A_{2}}$, and finding their difference ${\displaystyle d=A_{2}-A_{1}=-4}$
3) Since we have a value for d, we want to use the formula for ${\displaystyle A_{n}}$ that involves d.

Solution:

Step 1:
The formula for ${\displaystyle S_{n}}$ that involves a common difference, d, is the one we want. The other formula involves a common ratio, r. So we have to determine the value of n, ${\displaystyle A_{1}}$, and ${\displaystyle A_{n}}$
Step 2:
Now we determine ${\displaystyle A_{n}}$ by finding d. To do this we use the formula ${\displaystyle A_{n}=A_{1}+d(n-1)}$ with n = 2, ${\displaystyle A_{1}=27}$, and${\displaystyle A_{2}=23}$. This yields d = -4.
Step 3:
Now we have d, and we can use the same formula for ${\displaystyle A_{n}}$ to get ${\displaystyle A_{10}}$ and ${\displaystyle A_{15}}$. Using these formulas with the appropriate values will yield
${\displaystyle {\begin{array}{rcl}A_{15}&=&27+(-4)(15-1)\\&=&27-56\\&=&-39\end{array}}}$
and
${\displaystyle {\begin{array}{rcl}A_{10}&=&27+(-4)(10-1)\\&=&27-36\\&=&-9\end{array}}}$
Step 4:
Since we found ${\displaystyle A_{15}}$ in the last step, and we found the necessary pieces, we find ${\displaystyle S_{10}}$ by using the formula ${\displaystyle S_{10}={\frac {10}{2}}(27+-9)=5(-18)=-90}$
${\displaystyle {\begin{array}{rcl}A_{15}&=&27+(-4)(15-1)\\&=&27-56\\&=&-39\end{array}}}$
${\displaystyle S_{10}=-90,A_{15}=-39}$