An Amazing Series

Algebra Level 2

S = 1 + 2 + 3 + ⋯ + ( n − 1 ) + n + ( n − 1 ) + … + 3 + 2 + 1 S=1+2+3+\cdots+(n-1)+n+(n-1)+\ldots+3+2+1

Find general formula of S S .

n 2 {n}^{2} 2 n 2 2{n}^{2} ( n 2 ) 2 {\left(\dfrac{n}{2}\right)}^{2} 2 n 2n

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Dinesh Nath Goswami by Dinesh Nath Goswami , 20, India

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4 solutions

Michael Fuller
Dec 19, 2015

= ~~~~~\huge =~~

39 Helpful 0 Interesting 0 Brilliant 0 Confused

I like visual thinking

Denis Herda - 5 years, 5 months ago
Akshat Sharda
Dec 18, 2015

S = 1 + 2 + 3 + . . . + ( n − 1 ) + n + ( n − 1 ) + . . . + 3 + 2 + 1 = 2̸ ⋅ n ( n + 1 ) 2̸ − n = n 2 + n̸ − n̸ = n 2 \begin{aligned} S & = 1+2+3+...+(n-1)+n+(n-1)+...+3+2+1 \\ & = \not{2} \cdot \frac{n(n+1)}{\not{2}}-n= n^2+\not{n}-\not{n} \\ & = \boxed{n^2} \end{aligned}

25 Helpful 0 Interesting 0 Brilliant 0 Confused
Rishabh Jain
Dec 20, 2015

S=2(1+2+3...+n)-n =2(n/2(2+n-1))-n =n^2

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Ashish Menon
Jun 21, 2016

It reduces to 2 × n ( n + 1 ) 2 − n = n 2 + n − n = n 2 2 × \dfrac{n(n + 1)}{2} - n = n^2 + n - n = \color{#3D99F6}{\boxed{n^2}} .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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