Problem 13 · 2023 Math Kangaroo
Medium
Algebra & Patterns
substitutionsum-constraint
Consider the five numbers \(a_1, a_2, a_3, a_4, a_5\) with sum S. It is known that \(a_k = k + S\) for \(1 \le k \le 5\). What is the value of S?
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Answer: B — \(-\dfrac{15}{4}\)
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Hint 1 of 2
Add the five equations a_k = k + S together.
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Hint 2 of 2
The left side is just S again.
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Approach: sum all five relations
- Summing a_k = k + S over k = 1..5 gives S = (1+2+3+4+5) + 5S = 15 + 5S.
- So −4S = 15, giving S = −15/4.
- Hence S = −15/4.
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