In honor of Carl Gauss [No Calc!]

Algebra Level 1

What is

1 + 2 + 3 + 4 + 5 + ⋯ + 98 + 99 + 100 = ? \large1+2+3+4+5+\cdots+98+99+100=?

or

∑ n = 1 100 n = ? \large\sum_{n=1}^{100} n=?

Do not use a calculator, not even G o o g l e ! \text{Do not use a calculator, not even} \, \Large\color{#3D99F6}{\text{G}}\color{#D61F06}{\text{o}}\color{#CEBB00}{\text{o}}\color{#3D99F6}{\text{g}}\color{#20A900}{\text{l}}\color{#D61F06}{\text{e}}\color{#333333}{!}


The answer is 5050.

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5 solutions

Viki Zeta
Jul 14, 2016

Let sum to n terms be S. S n = 1 + 2 + 3 + … + n (+) S n = n + ( n − 1 ) + ( n − 2 ) + … + 1 ⟹ 2 S n = ( n + 1 ) + ( n − 1 + 2 ) + ( n − 2 + 3 ) + … + ( n + 1 ) ⟹ 2 S n = ( n + 1 ) + ( n + 1 ) + ( n + 1 ) + … + ( n + 1 ) ⟹ 2 S n = n 2 + n ( 1 ) ⟹ 2 S n = n ( n + 1 ) ⟹ S n = 1 2 n ( n + 1 ) (or) S = n ( n + 1 ) 2 On substituting n = 100, S 100 = 100 ( 100 + 1 ) 2 = 100 ( 101 ) 2 = 50 ( 101 ) ⟹ S 100 = 5050 \text{Let sum to n terms be S.} \\ \quad\quad S_n = 1 + 2 + 3 + \ldots + n \\ \text{(+) } S_n = n + (n-1) + (n-2) + \ldots + 1 \\ \implies 2S_n = (n+1) + (n-1+2) + (n-2+3) + \ldots + (n+1) \\ \implies 2S_n = (n+1) + (n+1) + (n+1) + \ldots + (n+1) \\ \implies 2S_n = n^2 + n(1)\\ \implies 2S_n = n(n+1) \\ \implies S_n = \frac{1}{2} n(n+1) \text{ (or) } S = \dfrac{n(n+1)}{2} \\ \text{On substituting n = 100, } \\ S_{100} = \dfrac{100(100+1)}{2} = \dfrac{100(101)}{2} = 50(101) \\ \implies S_{100} = \fbox{5050}

7 Helpful 0 Interesting 0 Brilliant 0 Confused

1+100=101 2+99=101 3+98=101 ...... ...... ..... 50+51=101

So 101×50=5050 Gauss wins

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Sanyam Goel
Jul 14, 2016

sum of 1 st 'n' natural numbers is given by n(n+1)/2...

so sum of 1 st 100 natural numbers

= 100*101/2= 5050.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Sn = n/2 {2a + (n-1)d} Just put n = 100 , a = 1 and d =1 . Surprisingly carl gauss did it when he was just 4

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Brittany Marie
Sep 16, 2016

10 1's + 10 2's + 10 3's, ect up until nine. Then 10 20's + 10 30's + 10 40's, etc until 90. So 10+20+30+40+50+60+70+80+90+100+200+300+400+500+600+700+800+900= 5050

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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