008A Sample Final A, Question 8

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

Foundations
1) Which of the $S_{n}$ 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?
Answer:
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 $A_{1}$ and $A_{2}$, and finding their difference $d=A_{2}-A_{1}=-4$
3) Since we have a value for d, we want to use the formula for $A_{n}$ that involves d.

Solution:

Step 1:
The formula for $S_{n}$ 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, $A_{1}$, and $A_{n}$

Step 2:
Now we determine $A_{n}$ by finding d. To do this we use the formula $A_{n}=A_{1}+d(n-1)$ with n = 2, $A_{1}=27$, and$A_{2}=23$. This yields d = -4.

Step 3:
Now we have d, and we can use the same formula for $A_{n}$ to get $A_{10}$ and $A_{15}$. Using these formulas with the appropriate values will yield
${\begin{array}{rcl}A_{15}&=&27+(-4)(15-1)\\&=&27-56\\&=&-39\end{array}}$
and
${\begin{array}{rcl}A_{10}&=&27+(-4)(10-1)\\&=&27-36\\&=&-9\end{array}}$

Step 4:
Since we found $A_{15}$ in the last step, and we found the necessary pieces, we find $S_{10}$ by using the formula $S_{10}={\frac {10}{2}}(27+-9)=5(-18)=-90$
${\begin{array}{rcl}A_{15}&=&27+(-4)(15-1)\\&=&27-56\\&=&-39\end{array}}$

Final Answer:
$S_{10}=-90,A_{15}=-39$

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