Mathematics - Sequence & Series Question with Solution | TestHub

The sum is ${\sum }_{i=1}^n(a+i-1)(b+i-1)={\sum }_{i=1}^n(ab+(a+b)(i-1)+(i-1{)}^2)$. Using ${\sum }_{i=1}^n(i-1)=\frac{n(n-1)}{2}$ and ${\sum }_{i=1}^n(i-1{)}^2=\frac{n(n-1)(2n-1)}{6}$, and given $ab=\frac{1}{6}$ and $a+b=\frac{1}{3}$,

the sum simplifies to $\frac{n}{6}(1-2n+2n^2)$.