MathAdmin: Created page with " Compute $\displaystyle{\sum_{n = 1}^6 4\left(\frac{1}{2}\right)^n}$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;" ! Foundati..."

2015-06-01T05:49:44Z

Created page with "<span class="exam"> Compute <math>\displaystyle{\sum_{n = 1}^6 4\left(\frac{1}{2}\right)^n}</math> {| class="mw-collapsible mw-collapsed" style = "text-align:left;" ! Foundati..."

New page

<span class="exam"> Compute <math>\displaystyle{\sum_{n = 1}^6 4\left(\frac{1}{2}\right)^n}</math>
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Foundations
|-
| What is the formula for the sum of the first n terms of a geometric sequence?
|-
|Answer:
|-
|The sum of the first n terms of a geometric sequence is <math> S_n=\frac{A_1(1-r^n)}{1-r}</math>
|-
|where <math>r</math> is the common ratio and <math>A_1</math> is the first term of the geometric sequence.
|}

Solution:

{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 1:
|-
|The common ratio for this geometric sequence is <math>r=\frac{1}{2}</math>.
|-
|The first term of the geometric sequence is <math>4\frac{1}{2}=2</math>.
|}

{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 2:
|-
|Using the above formula, we have
|-
|<math>\displaystyle{\sum_{n = 1}^6 4\left(\frac{1}{2}\right)^n}=S_6=\frac{2(1-(\frac{1}{2})^6)}{(1-\frac{1}{2})}</math>
|}

{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 3:
|-
|If we simplify, we get
|-
|<math>\displaystyle{\sum_{n = 1}^6 4\left(\frac{1}{2}\right)^n}=\frac{2(1-\frac{1}{64})}{\frac{1}{2}}=4\frac{63}{64}=\frac{63}{16}</math>.
|}

{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
|-
|<math>\frac{63}{16}</math>
|}

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

004 Sample Final A, Problem 13 - Revision history

MathAdmin: Created page with " Compute \displaystyle{\sum_{n = 1}^6 4\left(\frac{1}{2}\right)^n} {| class="mw-collapsible mw-collapsed" style = "text-align:left;" ! Foundati..."

MathAdmin: Created page with " Compute $\displaystyle{\sum_{n = 1}^6 4\left(\frac{1}{2}\right)^n}$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;" ! Foundati..."