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 $S_{n}={\frac {A_{1}(1-r^{n})}{1-r}}$
where $r$ is the common ratio and $A_{1}$ is the first term of the geometric sequence.

Solution:

Step 1:
The common ratio for this geometric sequence is $r={\frac {1}{2}}$.
The first term of the geometric sequence is $4{\frac {1}{2}}=2$.

Step 2:
Using the above formula, we have
$\displaystyle {\sum _{n=1}^{6}4\left(({\frac {1}{2}}\right))^{n}}=S_{6}={\frac {2(1-{\frac {1}{2}}^{6})}{1-{\frac {1}{2}}}}$

Step 3:

Step 4:

Final Answer:

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