005 Sample Final A, Question 21

Question Find the sum

$5+9+13+\cdots +49$

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) Take two adjacent terms in the sequence, say $A_{1}$ and $A_{2}$, and d = $A_{2}-A_{1}$
3) Since we have a value for d, we want to use the formula for $A_{n}$ that involves d.

Step 1:
This is the sum of an arithmetic sequence. The common difference is $d=4$. Since the formula for an arithmetic sequence is
$a_{n}=a_{1}+d(n-1)$, the formula for this arithmetic sequence is $a_{n}=5+4(n-1)$.

Step 2:
We need to figure out how many terms we are adding together. To do this, we let $a_{n}=49$ in the formula above and solve for $n$.

Step 3:
If $49=5+4(n-1)$, we have $44=4(n-1)$. Dividing by 4, we get $11=n-1$. Therefore, $n=12$.

Step 4:
The formula for the sum of the first n terms of an arithmetic sequence is $S_{n}={\frac {1}{2}}n(a_{1}+a_{n})$.

Step 5:
Since we are adding 12 terms together, we want to find $S_{12}$. So, $S_{12}={\frac {1}{2}}(12)(5+49)=324$.

Final Answer:
324