MathAdmin: Created page with "''' Question''' Find the sum
$5 + 9 + 13 + \cdots + 49$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Foundations |-..."

2015-06-01T01:39:14Z

Created page with "''' Question''' Find the sum <br> <center><math> 5 + 9 + 13 + \cdots + 49 </math></center> {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Foundations |-..."

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''' Question''' Find the sum <br>
<center><math> 5 + 9 + 13 + \cdots + 49 </math></center>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations
|-
|1) Which of the <math>S_n</math> 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:
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|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 <math>A_1</math> and <math>A_2</math>, and d = <math>A_2 - A_1</math>
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|3) Since we have a value for d, we want to use the formula for <math>A_n</math> that involves d.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! 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>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! 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>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! 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>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 4:
|-
|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>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! 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>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
|-
| 324
|}

[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]

005 Sample Final A, Question 21 - Revision history

MathAdmin: Created page with "''' Question''' Find the sum 5 + 9 + 13 + \cdots + 49 {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Foundations |-..."

MathAdmin: Created page with "''' Question''' Find the sum
$5 + 9 + 13 + \cdots + 49$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Foundations |-..."