008A Sample Final A, Question 8

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Question: Given a sequence use formulae to compute and .

Foundations
1) Which of the 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 and , and finding their difference
3) Since we have a value for d, we want to use the formula for that involves d.

Solution:

Step 1:
The formula for 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, , and
Step 2:
Now we determine by finding d. To do this we use the formula with n = 2, , and. This yields d = -4.
Step 3:
Now we have d, and we can use the same formula for to get and . Using these formulas with the appropriate values will yield
and
Step 4:
Since we found in the last step, and we found the necessary pieces, we find by using the formula

Final Answer:

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