Question

The difference between the sum of the first $k$ terms of the series $1^3+2^3+3^3+\dots \dots +n^3$ and the sum of the first $k$ terms of $1+2+3+\dots \dots n$ is $1980$. The value of $k$ is

Answer 1

${\left[\frac{k(k+1)}{2}\right]}^2-\frac{k(k+1)}{2}=1980$
$\Rightarrow \frac{k(k+1)}{2}[\frac{k(k+1)}{2}-1]=1980$
$\Rightarrow k(k+1)(k^2+k-2)=1980\times 4$
$\Rightarrow (k-1)k(k+1)(k+2)=8\cdot 9\cdot 10\cdot 11$
$\therefore k-1=8\Rightarrow k=9$