Difference between revisions of "1971 AHSME Problems/Problem 33"
(solution that doesn't rely on answer choices, see also box) |
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+ | == 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 <math>\text{meters}</math>. Then, <math>S</math> should have units of <math>\text{meters}</math>, <math>S^{\prime}</math> units of <math>\tfrac1{\text{meters}}</math> and <math>P</math> units of <math>\text{meters}^n</math>. Therefore, <math>SS^{\prime}</math> is unitless, so we can eliminate options (A) and (C). <math>S/S^{\prime}</math> has units <math>\text{meters}^2</math>, so, to equal <math>P</math> (which has units <math>\text{meters}^n</math>), the exponent needs to be <math>\tfrac n2</math>. The only remaining answer choice which satsifies this constraint is <math>\boxed{\textbf{(B) }(S/S^{\prime})^{\frac12n}}</math>. | ||
== See Also == | == See Also == |
Latest revision as of 17:23, 8 August 2024
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 | ||
All AHSME Problems and Solutions |
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