Evidence . Sum of n+(n+1)+...+(n+10) is no primenumber - induction

Question

My task is: Show that the sum s(n)=n+(n+1)+(n+2)+...+(n+10) is no prime number.

I think, I can solve the problem with induction, but thats all I know. I either dont unterstand induction so please, can anybody help me?

Thanks for every hint and help!!

Answer 1

Simplify your sum : you get $$n+(n+1)+...+(n+10)=11n+55=11\times(n+5)$$ and a such number is divisible by $11$ and so is not a prime.

Answer 2

$s(n+1)-s(n)=(n+1)+(n+2)+\dots+(n+11) - n - (n+1) - \dots - (n+10) = (n+11)-n=11$

$s(0)=55$ and $11|55$.

So $11|s(n)\to 11|s(n+1)$ and as this is true for $s(0)$, it is true for all $n\ge0$.

Answer 3

Note that the sum $1+2+3+...+10$ can be thought of as 5 pairs of numbers which add up to $11$, i.e. $10+1, 9+2, 8+3, 7+4, 5+6$. Since there are 5 of these this sum will be 55. Note that we are adding our number $n$ to itself $11$ times will also yield $11n$. Therefore our sum will always be of the form $11n+55=11(n+5)$. Since our sum will always be divisible by $11$, we know that this sum will not be prime.

Hopefully this makes sense :)