Question:
If \( f(0) = 0, f(1) = 1, f(2) = 2 \) and \( f(x) = f(x-2) + f(x-3) \) for \( x = 3, 4, 5, \dots \), then find \( f(10) \).
If \( f(0) = 0, f(1) = 1, f(2) = 2 \) and \( f(x) = f(x-2) + f(x-3) \) for \( x = 3, 4, 5, \dots \), then find \( f(10) \).
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To solve recurrence relations, start by substituting known values and continue using the relation to calculate the next terms.
Updated On: May 13, 2025
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Solution and Explanation
Given the function relation:
\[ f(x) = f(x-2) + f(x-3) \] We know the values for \( f(0) = 0, f(1) = 1, f(2) = 2 \). We can calculate the subsequent values for \( f(x) \) using the recurrence relation. - \( f(3) = f(1) + f(0) = 1 + 0 = 1 \) - \( f(4) = f(2) + f(1) = 2 + 1 = 3 \) - \( f(5) = f(3) + f(2) = 1 + 2 = 3 \) - \( f(6) = f(4) + f(3) = 3 + 1 = 4 \) - \( f(7) = f(5) + f(4) = 3 + 3 = 6 \) - \( f(8) = f(6) + f(5) = 4 + 3 = 7 \) - \( f(9) = f(7) + f(6) = 6 + 4 = 10 \) - \( f(10) = f(8) + f(7) = 7 + 6 = 13 \) Thus, \( f(10) = 13 \).
\[ f(x) = f(x-2) + f(x-3) \] We know the values for \( f(0) = 0, f(1) = 1, f(2) = 2 \). We can calculate the subsequent values for \( f(x) \) using the recurrence relation. - \( f(3) = f(1) + f(0) = 1 + 0 = 1 \) - \( f(4) = f(2) + f(1) = 2 + 1 = 3 \) - \( f(5) = f(3) + f(2) = 1 + 2 = 3 \) - \( f(6) = f(4) + f(3) = 3 + 1 = 4 \) - \( f(7) = f(5) + f(4) = 3 + 3 = 6 \) - \( f(8) = f(6) + f(5) = 4 + 3 = 7 \) - \( f(9) = f(7) + f(6) = 6 + 4 = 10 \) - \( f(10) = f(8) + f(7) = 7 + 6 = 13 \) Thus, \( f(10) = 13 \).
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