Question

The sum of consecutive integers 1,2,3,...,n is given by the formula 1/2n(n+1). How many consecutive integers, starting with 1, must be added to get the sum of 1081

Answer 1

$\dfrac{1}{2}n(n+1)=1081\qquad|\text{multiply both sides by 2}\\\\n(n+1)=2162\qquad|\text{use distributive property}\\\\(n)(n)+(n)(1)=2162\qquad|\text{subtract 2162 from both sides}\\\\n^2+n-2162=0\\\\n^2+47n-46n-(46)(47)=0\\\\n(n+47)-46(n+47)=0\\\\(n+47)(n-46)=0\iff n+47=0\ \vee\ n-46=0\\\\n=-47\notin\mathbb{N}\ \vee\ n=46\\\\Answer:\ 46\ integers.$