Problem 24 · 2018 Math Kangaroo
Stretch
Algebra & Patterns
substitutionevaluate-formula
A function \(f\) fulfils the property \(f(x + y) = f(x) \cdot f(y)\) for all whole numbers \(x\) and \(y\). Furthermore \(f(1) = \tfrac{1}{2}\). Determine the value of the expression \(f(0) + f(1) + f(2) + f(3)\).
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Answer: D — \(\tfrac{15}{8}\)
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Hint 1 of 2
Plug in x = y = 0 to pin down f(0).
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Hint 2 of 2
Build f(2) and f(3) by repeatedly using f(a+b) = f(a)f(b).
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Approach: exploit the multiplicative functional equation
- Setting x = y = 0 gives f(0) = f(0)², so f(0) = 1.
- f(2) = f(1)² = 1/4 and f(3) = f(1)³ = 1/8.
- Sum: 1 + 1/2 + 1/4 + 1/8 = 15/8.
- Answer: 15/8.
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