10 times n

Number Theory Level 1

For which positive integer $n$ do we have

$1 + 2 + 3 + 4 + \ldots + n = 10 \times n ?$

The answer is 19.

7 solutions

Ryan Soedjak
Dec 28, 2013

$\frac{n(n+1)}2=10n\implies n=19.$

will u plz tell me how it is

yamini latha - 7 years, 5 months ago

The sum of $n$ consecutive integers (from $1$ to $n$ ) is equal to $\frac{n(n+1)}{2}$ . Therefore, we have, $\frac{n(n+1)}{2} = 10n \Rightarrow n^2 + n =20n \Rightarrow n^2-19n=0$ Solving gives $n=0, 19$ however $n$ is a positive integer so $n=\fbox{19}$

Oliver Welsh - 7 years, 5 months ago

Dang Anh Tu
Jun 9, 2014

Please read this note, you will understand:

https://brilliant.org/discussions/thread/discuss-more-about-pyramid-investigations-by-arron/

Partha Ghosh
Apr 2, 2014

n(n+1)/2 =10n => n=19 (ans)

Sunil Pradhan
Mar 16, 2014

1 + 2 + 3 + ... + n = 10 × n n(n+1)/2 = 10n (n+1)/2 = 10 n + 1 = 20

n = 19

Prasun Biswas
Feb 22, 2014

We should first know about the identity for the series : $1+2+3+....+n=\frac{n(n+1)}{2}$ . So, on applying this in the problem, we get ----

$1+2+3+......+n=10\times n$

$\implies \frac{n(n+1)}{2}=10n$

$\implies \frac{n+1}{2}=10 \implies n+1=20 \implies n=\boxed{19}$

Josh Grotstein
Feb 18, 2014

The sum of all integers from 1 to N = ((N+1)/2)*N

If we set that = to 10 * N, then...

(N+1)/2 = 10, and

N+1 = 20, and

N =19

Mesbah Tanvir
Dec 29, 2013

n(n+1)/2=10n

just solve it ... ans 19

10 times n

The answer is 19.

7 solutions

n = 19

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