Compute 
| 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{n}={\frac {A_{1}(1-r^{n})}{1-r}}}
|
| where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}
is the common ratio and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1}
is the first term of the geometric sequence.
|
Solution:
| Step 1:
|
| The common ratio for this geometric sequence is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=\frac{1}{2}}
.
|
| The first term of the geometric sequence is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\frac{1}{2}=2}
.
|
| Step 2:
|
| Using the above formula, we have
|
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}}
|
Return to Sample Exam