Undetermined Coefficients (Part 3)

Algebra Level 3

∑ n = 1 50 ( 2 n − 1 ) ( n + 2 ) 2 = 1 ⋅ 3 2 + 3 ⋅ 4 2 + 5 ⋅ 5 2 + ⋯ + 99 ⋅ 52 2 = ? \sum _{ n=1 }^{ 50 }{ \left( 2n-1 \right) { \left( n+2 \right) }^{ 2 } } =1\cdot { 3 }^{ 2 }+3\cdot { 4 }^{ 2 }+5\cdot { 5 }^{ 2 }+\cdots +99\cdot { 52 }^{ 2 } = \, ?

You may want to read the wiki Method of Undetermined Coefficients .


The answer is 3556625.

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Arulx Z by Arulx Z , 20, India

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2 solutions

Priyanshu Mishra
Dec 8, 2017

We have

∑ n = 1 50 ( 2 n − 1 ) ( n + 2 ) 2 = ∑ n = 1 50 ( 2 n 3 + 7 n 2 + 4 n − 4 ) \large\ \sum _{ n=1 }^{ 50 }{ \left( 2n - 1 \right) { \left( n + 2 \right) }^{ 2 } } = \sum _{ n=1 }^{ 50 }{ \left( 2{ n }^{ 3 } + 7{ n }^{ 2 } + 4n - 4 \right) }

which equals 3556625 \boxed{3556625}

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Clyde Castial
Nov 27, 2019

just use the " Method of Undetermined Coefficients," or in other term "mathematical induction"

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