Question

Find the sum of $n-$terms of the series $1.2+2.3+3.4+...$

Answer 1

Let $T_r$ be the general term of the series
S0 $T_r=r(r+1)$
To express $t_r=f\left(r\right)-f(r-1)$ multiply and divide $t_r$ by $[(r+2)-(r-1)]$
so $T_r=\frac{r}{3}(r+1)[(r+2)-(r-1)]=\frac{1}{3}[r(r+1)(r+2)-(r-1)r(r+1)]$
Let $f\left(r\right)=\frac{1}{3}r(r+1)(r+2)$
so $T_r=[f\left(r\right)-f(r-1)]$
Now $S={\sum }_{r=1}^n{T}_r=T_1+T_2+T_3+...+T_n$
$T_1=\frac{1}{3}[\mathrm{1.2.3}-0]$
$T_2=\frac{1}{3}[\mathrm{2.3.4}-\mathrm{1.2.3}]$
$T_3=\frac{1}{3}[\mathrm{3.4.5.}-\mathrm{2.3.4}]$
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$T_n=\frac{1}{3}[n(n+1)(n+2)-(n-1)n(n+1)]$
$S=\frac{1}{3}n(n+1)(n+2)$