004 Sample Final A, Problem 13

Compute $\displaystyle {\sum _{n=1}^{6}4\left({\frac {1}{2}}\right)^{n}}$

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 (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 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}}}

Step 3:

Step 4:

Final Answer:

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