4.7 Q-3

Exercise 4.7 — Question 3

This question involves arithmetic-geometric sequences and/or summation formulas for natural numbers.

Key Formulas

Sum of first $n$ natural numbers: ${\sum }_{k=1}^{n}k=\frac{n(n+1)}{2}$

Sum of squares of first $n$ natural numbers: ${\sum }_{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}$

Sum of cubes of first $n$ natural numbers: ${\sum }_{k=1}^n{k}^3={\left[\frac{n(n+1)}{2}\right]}^2$

General term of an arithmetic-geometric sequence:

If $\{a_n\}$ is arithmetic with first term $a$ and common difference $d$, and $\{b_n\}$ is geometric with common ratio $r$, then the arithmetic-geometric sequence has general term: $T_n=[a+(n-1)d]r^{n-1}$

Sum to $n$ terms of an arithmetic-geometric series (using the subtraction 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 $|r|<1$): $S_{\infty }=\frac{a}{1-r}+\frac{dr}{(1-r{)}^2}$

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 $T_n$ using the formula above.
3. Apply the appropriate summation formula.
4. Use sigma notation to express the series compactly.