005 Sample Final A, Question 21

Question Find the sum

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

Step 1:
This is the sum of an arithmetic sequence. The common difference is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 49=5+4(n-1)} , we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 44=4(n-1)} . Dividing by 4, we get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 11=n-1} . Therefore, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{12}} . So, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{12}={\frac {1}{2}}(12)(5+49)=324} .

Final Answer:
324