Crazy Numbers

Algebra Level 5

1 4 − 2 4 + 3 4 − 4 4 + ⋯ − 5 0 4 + 5 1 4 = ? 1^4 - 2^4 + 3^4 - 4^4 +\cdots - 50^4 + 51^4=\ ? \

This problem is inspired by many posts of similar kind.


The answer is 3515226.

Dev Sharma by Dev Sharma , 20, India

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1 solution

Akshat Sharda
Nov 23, 2015

∑ n = 0 25 ( 2 n + 1 ) 4 − ∑ n = 1 25 ( 2 n ) 4 1 4 + ∑ n = 1 25 ( 16 n 4 + 32 n 3 + 24 n 2 + 8 n + 1 ) − ∑ n = 1 25 ( 16 n 4 ) 1 + 16 ∑ n = 1 25 n 4 + ∑ n = 1 25 32 n 3 + 24 ∑ n = 1 25 n 2 + 8 ∑ n = 1 25 n + ∑ n = 1 25 1 − 16 ∑ n = 1 25 n 4 1 + 32 ⋅ ( 25 × 26 2 ) 2 + 24 ⋅ 25 × 26 × 51 6 + 8 ⋅ 25 × 26 2 + 25 = 3515226 \displaystyle \sum^{25}_{n=0}(2n+1)^4-\displaystyle \sum^{25}_{n=1}(2n)^4 \\ 1^4+\displaystyle \sum^{25}_{n=1}(16n^4+32n^3+24n^2+8n+1)-\displaystyle \sum^{25}_{n=1}(16n^4) \\ 1+\cancel{16\displaystyle \sum^{25}_{n=1}n^4}+\displaystyle \sum^{25}_{n=1}32n^3+24\displaystyle \sum^{25}_{n=1}n^2+8\displaystyle \sum^{25}_{n=1}n+\displaystyle \sum^{25}_{n=1}1-\cancel{16\displaystyle \sum^{25}_{n=1}n^4} \\ 1+32\cdot \left(\frac{25\times26}{2}\right)^2+24\cdot \frac{25\times 26 \times 51}{6}+8 \cdot \frac{25 \times 26}{2}+25=\boxed{3515226}

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Moderator note:

Simple standard approach once you know how to deal with sum of powers of integers.

Exactly!!

Even your solutions inspired me to made this problem...

Dev Sharma - 5 years, 6 months ago

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