Difference between revisions of "005 Sample Final A, Question 21"

Revision as of 12:50, 20 May 2015

Question Find the sum

5+9+13+\cdots +49

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

@@ Line 5: / Line 5: @@
 ! Step 1:
 |-
-|
+|This is the sum of an arithmetic sequence. The common difference is <math>d=4</math>. Since the formula for an arithmetic sequence is
+|-
+|<math>a_n=a_1+d(n-1)</math>, the formula for this arithmetic sequence is <math>a_n=5+4(n-1)</math>.
 |}
@@ Line 11: / Line 13: @@
 ! Step 2:
 |-
-|
+| We need to figure out how many terms we are adding together. To do this, we let <math>a_n=49</math> in the formula above and solve for <math>n</math>.
 |}
@@ Line 17: / Line 19: @@
 ! Step 3:
 |-
-|
+| If <math>49=5+4(n-1)</math>, we have <math>44=4(n-1)</math>. Dividing by 4, we get <math>11=n-1</math>. Therefore, <math>n=12</math>.
-|-
-|
 |}
@@ Line 25: / Line 25: @@
 ! Step 4:
 |-
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+|The formula for the sum of the first n terms of an arithmetic sequence is <math>S_n=\frac{1}{2}n(a_1+a_n)</math>.
 |}
@@ Line 31: / Line 31: @@
 ! Step 5:
 |-
-|
+|Since we are adding 12 terms together, we want to find <math>S_{12}</math>. So, <math>S_{12}=\frac{1}{2}(12)(5+49)=324</math>.
-|-
-|
-|-
-|
 |}
@@ Line 41: / Line 37: @@
 ! Final Answer:
 |-
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+| 324
 |}