1969 AHSME Problems/Problem 33
Problem
Let and be the respective sums of the first terms of two arithmetic series. If for all , the ratio of the eleventh term of the first series to the eleventh term of the second series is:
Solution
Let be the first arithmetic sequence and be the second arithmetic sequence. If , then . Since and are just the first term, the first term of is and the first term of is for some . If , then , so the sum of the first two terms of is and the sum of the first two terms of is for some . Thus, the second term of is and the second term of is , so the common difference of is and the common difference of is .
Thus, using the first terms and common differences, the sum of the first three terms of equals , and the sum of the first three terms of equals . That means With the substitution, the common difference of is , and the common difference of is . That means the term of is , and the term of is . Thus, the ratio of the eleventh term of the first series to the eleventh term of the second series is .
Solution 2 (Quick Sol)
Note that the sum of arithmetic sequences can be expressed as a quadratic polynomial of . From the ratio given in the problem, multiply to both sides to get quadratic polynomials. From there, the 11th term for and can becalculated. The ratio is .
See Also
1969 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 32 |
Followed by Problem 34 | |
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