4.7 Q-1

Exercise 4.7 — Question 1

Arithmetic-Geometric Sequence: Key Concepts

An arithmetic-geometric sequence is formed by multiplying the corresponding terms of an arithmetic sequence and a geometric sequence.

If the arithmetic sequence has first term $a$ and common difference $d$, and the geometric sequence has first term $1$ and common ratio $r$, then the arithmetic-geometric sequence is:

$a,(a+d)r,(a+2d)r^2,(a+3d)r^3,\dots$

General Term

The $n$-th term (general term) of an arithmetic-geometric sequence is:

$T_n=[a+(n-1)d]r^{n-1}$

Sum to $n$ Terms

To find $S_n$, use the subtraction method:

1. Write $S_n=a+(a+d)r+(a+2d)r^2+\cdots +[a+(n-1)d]r^{n-1}$
2. Multiply both sides by $r$: $r{S}_n=ar+(a+d)r^2+\cdots +[a+(n-2)d]r^{n-1}+[a+(n-1)d]r^n$
3. Subtract: $(1-r)S_n=a+d(r+r^2+\cdots +r^{n-1})-[a+(n-1)d]r^n$
4. Simplify using the geometric series formula.

$S_n=\frac{a-[a+(n-1)d]r^n}{1-r}+\frac{dr(1-r^{n-1})}{(1-r{)}^2},r\ne 1$

Sum to Infinity

When $|r|<1$, as $n\to \infty$, $r^n\to 0$, so:

$S_{\infty }=\frac{a}{1-r}+\frac{dr}{(1-r{)}^2}$

Sigma Notation

The series $a+(a+d)r+(a+2d)r^2+\cdots$ can be written as:

${\sum }_{k=1}^n[a+(k-1)d]r^{k-1}$

For example, the series $1+2r+3r^2+4r^3+\cdots$ (where $a=1$, $d=1$) is:

${\sum }_{k=1}^{\infty }k{r}^{k-1}$