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 $\displaystyle {\sum _{k=1}^{\infty }a_{k}}$

Foundations
1) What type of series is this?
2) Which formulas, about this type of series, are relevant to this question?
3) In the formula there are some placeholder variables. What is the value of each placeholder?
Answer:
1) This series is geometric. The giveaway is there is a number raised to the nth power.
2) The desired formulas are Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{n}=a\cdot r^{n-1}}   and   $S_{\infty }={\frac {a_{1}}{1-r}}$
3) $a_{1}$ is the first term in the series, which is $-3$. The value for r is the ratio between consecutive terms, which is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {-1}{3}}}

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{n}=ar^{n-1}} where $a$ is the first term of the sequence.
So, the formula for this geometric series is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{n}=(-3)\left({\frac {-1}{3}}\right)^{n-1}} .

Step 3:
For geometric series, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \displaystyle {\sum _{k=1}^{\infty }a_{k}}={\frac {a}{1-r}}} if $|r|<1$. Since $|r|={\frac {1}{3}}$,
we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \displaystyle {\sum _{k=1}^{\infty }a_{k}}={\frac {-3}{1-{\frac {-1}{3}}}}={\frac {-9}{4}}} .

Final Answer:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{n}=(-3)\left({\frac {-1}{3}}\right)^{n-1}}
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 \frac{-9}{4}}

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