005 Sample Final A, Question 22

Question Consider the following sequence,

$-3,1,-{\frac {1}{3}},{\frac {1}{9}},-{\frac {1}{27}},\cdots$

     a. Determine a formula for $a_{n}$, the n-th term of the sequence.
     b. Find the sum $\displaystyle {\sum _{k=1}^{\infty }a_{k}}$

Step 1:
The sequence is a geometric sequence. The common ratio is $r={\frac {-1}{3}}$.

Step 2:
The formula for the nth term of a geometric series is $a_{n}=ar^{n-1}$ where $a$ is the first term of the sequence.
So, the formula for this geometric series is $a_{n}=(-3)\left({\frac {-1}{3}}\right)^{n-1}$.

Step 3:
For geometric series, $\displaystyle {\sum _{k=1}^{\infty }a_{k}}={\frac {a}{1-r}}$ if $|r|<1$. Since $|r|={\frac {1}{3}}$,
we have $\displaystyle {\sum _{k=1}^{\infty }a_{k}}={\frac {-3}{1-{\frac {-1}{3}}}}={\frac {-9}{4}}$.

Final Answer:
$a_{n}=(-3)\left({\frac {-1}{3}}\right)^{n-1}$
${\frac {-9}{4}}$