The very end

Algebra Level 5

Let f ( x ) f(x) be a quintic ( 5 th 5^\text{th} degree) polynomial with f ( n ) = 3 n + 1 f(n)=3^n+1 for all integers 1 ≤ n ≤ 6 1\leq n\leq 6 . Find f ( 8 ) f(8) .


The answer is 4834.

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Akshat Sharda by Akshat Sharda , 21, India

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1 solution

Otto Bretscher
Feb 28, 2016

f ( x ) = 1 + 3 ∑ k = 0 5 2 k ( x − 1 k ) f(x)=1+3\sum_{k=0}^52^k{x-1 \choose k} so f ( 8 ) = 1 + 3 ∑ k = 0 5 2 k ( 7 k ) = 1 + 3 ( 3 7 − 2 7 − 7 × 2 6 ) = 4834 f(8)=1+3\sum_{k=0}^5 2^k{7 \choose k}=1+3(3^7-2^7-7\times2^6)=\boxed{4834}

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