1971 AHSME Problems/Problem 33
Contents
[hide]Problem
If is the product of quantities in Geometric Progression, their sum, and the sum of their reciprocals, then in terms of , and is
Solution 1
Let the geometric sequence have first term and common ratio . Then, the first terms of the sequence are . The product of these terms is by the formula for triangular numbers. Using the sum formula reveals that .
We know that Combining fractions reveals that . Note that this denominator looks suspiciously similar to our formula for . In fact, . Because , our answer is .
Solution 2 (Answer Choices)
We can just look at a very specific case: Here, and
Then, plug in values of and into each of the answer choices and see if it matches the product.
Answer choice works:
-edited by coolmath34
Solution 3 (Answer Choices)
We can use dimensional analysis to cut down our answer choices. Suppose that each of the terms in the geometric progression is in units of . Then, should have units of , units of and units of . Therefore, is unitless, so we can eliminate options (A) and (C). has units , so, to equal (which has units ), the exponent needs to be . The only remaining answer choice which satsifies this constraint is .
See Also
1971 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 32 |
Followed by Problem 34 | |
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