008A Sample Final A, Question 8

Question: Given a sequence $27,23,19,15,\ldots$ use formulae to compute $S_{10}$ and $A_{15}$.

Foundations
1) Which of the $S_{n}$ formulas should you use?
2) What is the common ratio or difference?
3) How do you find the values you need to use the formula?
Answer:
1) The variables in the formulae give a bit of a hint. The r stands for ratio, and ratios are associated to geometric series. This sequence is arithmetic, so we want the formula that does not involve r.
2) We determine the common difference by taking two adjacent terms in the sequence, say $A_{1}$ 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_2} , and finding their difference 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 d = A_2 - A_1 = -4}
3) Since we have a value for d, we want to use the formula for 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_n} that involves d.

Solution:

Step 1:

The formula for 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} that involves a common difference, d, is the one we want. The other formula involves a common ratio, r. So we have to determine the value of n, 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} , 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_n}

Step 2:

Now we determine 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_n} by finding d. To do this we use the formula 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_n = A_1 + d(n - 1)} with n = 2, 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 = 27} , andFailed 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_2 = 23} . This yields d = -4.

Step 3:
Now we have d, and we can use the same formula for 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_n} to get 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_{10}} and $A_{15}$. Using these formulas with the appropriate values will yield
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}A_{15}&=&27+(-4)(15-1)\\&=&27-56\\&=&-39\end{array}}}
and
${\begin{array}{rcl}A_{10}&=&27+(-4)(10-1)\\&=&27-36\\&=&-9\end{array}}$

Step 4:
Since we found $A_{15}$ in the last step, and we found the necessary pieces, we find $S_{10}$ by using the formula $S_{10}={\frac {10}{2}}(27+-9)=5(-18)=-90$
${\begin{array}{rcl}A_{15}&=&27+(-4)(15-1)\\&=&27-56\\&=&-39\end{array}}$

Final Answer:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S_{10}=-90,A_{15}=-39}

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