4.7 Q-4

Exercise 4.7 — Question 4

This question involves arithmetic-geometric sequences and/or summation formulas (∑n, ∑n², ∑n³).

Key Formulas

Arithmetic-Geometric Sequence: A sequence formed by multiplying corresponding terms of an arithmetic sequence and a geometric sequence.

General term: $T_n=[a+(n-1)d]r^{n-1}$ where $a$ is the first term of the arithmetic part, $d$ is the common difference, and $r$ is the common ratio.

Sum to $n$ terms (multiply-and-subtract method): $S_n=\frac{a-[a+(n-1)d]r^n}{1-r}+\frac{dr(1-r^{n-1})}{(1-r{)}^2},r\ne 1$

Sum to infinity (when $|r|<1$): $S_{\infty }=\frac{a}{1-r}+\frac{dr}{(1-r{)}^2}$

Standard Summation Formulas: ${\sum }_{k=1}^{n}k=\frac{n(n+1)}{2}$ ${\sum }_{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}$ ${\sum }_{k=1}^n{k}^3={\left[\frac{n(n+1)}{2}\right]}^2$

Approach

1. Identify whether the series is arithmetic-geometric by checking if each term is a product of an arithmetic term and a geometric term.
2. Write the general term $T_n$ in the form $[a+(n-1)d]r^{n-1}$.
3. Apply the appropriate sum formula (finite or infinite).
4. For pure summation problems, apply the standard ∑n, ∑n², or ∑n³ formulas directly.