Animated Solution for Mathematics - Sequence and Series: If $S(x)=(1+x)+2(1+x{)}^2+3(1+x{)}^3+\cdots +60(1+x{)}^{60},x\ne 0$, and $(60{)}^{2}S(60)=a(b{)}^b+b$, where $a,b\in \mathbb{N}$, then $(a+b)$ equal to

The given series is $S(x)=(1+x)+2(1+x{)}^2+3(1+x{)}^3+\cdots +60(1+x{)}^{60}$.

At first glance, this expression appears complex. However, notice that the coefficients $1,2,3,\dots ,60$ form an arithmetic progression, while the terms $(1+x),(1+x{)}^2,\dots ,(1+x{)}^{60}$ form a geometric progression.

This structure identifies the series as an Arithmetico-Geometric Progression (AGP).

To simplify the algebra, let $y=1+x$. The series now takes the standard form:

This substitution removes the clutter and allows us to focus on the underlying algebraic structure of the series.

Shift and Subtract

To find the sum, we multiply the series $S$ by the common ratio $y$:

Now, we subtract the second equation from the first ($S-yS$):

This simplifies to a geometric progression minus the final term:

The sum of the geometric progression $y+y^2+\cdots +y^{60}$ is given by the formula $\frac{y(y^{60}-1)}{y-1}$. Substituting this into our equation:

Recalling that $y=1+x$, we have $1-y=-x$ and $y-1=x$. Substituting these back:

To evaluate at $x=60$ (where $y=61$), we multiply by $-60$:

Multiplying the entire equation by $-60$ to isolate $(60{)}^{2}S(60)$:

Expanding the terms:

Grouping the $61^{61}$ terms yields:

Comparing this to the form $a(b{)}^b+b$, we identify $a=3599$ and $b=61$. The final result is: