What does this add up to?

Algebra Level 2

1 + 2 + 3 + ⋯ + 100000 = ? 1+2+3+\cdots+100 000 = \, ?


The answer is 5000050000.

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Joel Casero by Joel Casero , 26, Philippines

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3 solutions

Arulx Z
Feb 26, 2016

1 + 2 + 3 + ⋯ + n = n ( n + 1 ) 2 1+2+3+\dots +n=\frac { n\left( n+1 \right) }{ 2 }

On plugging 100000 into n n ,

1 + 2 + 3 + ⋯ + 100000 = 100000 ( 100001 ) 2 = 5000050000 1+2+3+\dots +100000=\frac { 100000\left( 100001 \right) }{ 2 } =5000050000

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Moderator note:

How do we know the first equation is true?

Nice solution

Joel Casero - 5 years, 3 months ago

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Thanks a lot :)

Arulx Z - 5 years, 3 months ago

How do we know the first equation is true?

Calvin Lin Staff - 5 years, 3 months ago

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S = 1 + 2 + 3 + … + n S = n + ( n − 1 ) + ( n − 2 ) + … + 1 2 S = ( n + 1 ) + ( n + 1 ) + ( n + 1 ) + … + ( n + 1 ) \begin{matrix} S & = & 1 & + & 2 & + & 3 & + & \dots & + & n \\ S & = & n & + & \left( n-1 \right) & + & \left( n-2 \right) & + & \dots & + & 1 \\ 2S & = & \left( n+1 \right) & + & \left( n+1 \right) & + & \left( n+1 \right) & + & \dots & + & \left( n+1 \right) \end{matrix}

2 S = n ( n + 1 ) S = n ( n + 1 ) 2 2S=n\left( n+1 \right) \\ S=\frac { n\left( n+1 \right) }{ 2 }

Arulx Z - 5 years, 3 months ago
Joel Casero
Feb 24, 2016

50 000[1+100 000] = 50 000[100 001] = 5 000 050 000

1 Helpful 0 Interesting 0 Brilliant 0 Confused

It is an arithmetic progression with d = 1 , n = 100000 , a 1 = 1 d=1, n=100000, a_1=1 and a n = 100000 a_n=100000 .

So the sum of terms is

s = n 2 ( a 1 + a n ) = 100000 2 ( 1 + 100000 ) = 5000050000 s=\dfrac{n}{2}(a_1+a_n)=\dfrac{100000}{2}(1+100000)=\large{\color{#69047E}\boxed{5000050000}}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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