Exercise 4.6 — Question 4

Relevant Concepts

Exercise 4.6 typically covers arithmetic-geometric sequences, sigma notation, and special sums (∑n, ∑n², ∑n³). The following formulas are essential:

Special Sum Formulas

${\sum }_{k=1}^{n}k=\frac{n(n+1)}{2}$

${\sum }_{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}$

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

Arithmetic-Geometric Series

An arithmetic-geometric sequence is formed by multiplying 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 general term is:

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

Sum to $n$ terms (for $r\ne 1$):

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

Sum to infinity (when $|r|<1$):

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

Solution

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Refer to the FBISE Grade 11 Mathematics textbook, Chapter 4, Exercise 4.6, Question 4 for the complete problem.