A 10sec problem

Algebra Level 2

Solve for n.

∑ k = 3 n 2 − 3 k n 2 = 22 \sum_{k=3}^{n^2-3} \frac{k}{n^2}=22


The answer is 7.

Pieter-Jan Meuris by Pieter-Jan Meuris , 22, Belgium

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1 solution

∑ k = 3 n 2 − 3 k n 2 = 22 \sum_{k=3}^{n^2-3} \frac{k}{n^2}=22

⇔ 22 n 2 = ∑ l = 3 n 2 − 3 \Leftrightarrow 22n^2=\sum_{l=3}^{n^2-3}

⇔ 22 n 2 = 1 2 ( n 2 − 3 ) ( n 2 − 2 ) − 3 \Leftrightarrow 22n^2=\frac{1}{2}(n^2-3)(n^2-2)-3

⇔ 44 n 2 = n 4 − 5 n 2 \Leftrightarrow 44n^2=n^4-5n^2

⇔ n = 7 \Leftrightarrow n=7

2 Helpful 0 Interesting 0 Brilliant 0 Confused

It took me like 15 seconds.

Adrian Neacșu - 7 years, 1 month ago

I couldn't understand transition from step 2 to 3

Shuvam Nayak - 7 years, 1 month ago

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Mr Meuris applied a formula for the sum of terms each of which is just 'k'. He replaced the beginning and ending values as appropriate.

Bill Bell - 6 years, 11 months ago

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