Question

Sol. [Key Concept: Partial sums of a G.P. with first term $a$, ratio $r$.]

(B) $\frac{1}{a^{2}\left( 1 - r^{2} \right)}\left\lbrack 1 + \frac{1}{r^{2}} + \frac{1}{r^{4}} + \ldots\frac{1}{r^{2(n - 2)}} \right\rbrack = \frac{1 - \frac{1}{r^{2(n - 1)}}}{\left( 1 - \frac{1}{r^{2}} \right)a^{2}\left( 1 - r^{2} \right)}$

$$= \frac{\left( 1 - r^{2n} \right)}{a^{2}\left( 1 - r^{2} \right)^{2}r^{2n - 4}}$$

(C) $\frac{1}{a^{m}\left( 1 + r^{m} \right)}\left\lbrack 1 + \frac{1}{r^{m}} + \frac{1}{r^{2\text{ }m}} + \ldots + \frac{1}{r^{(n - 2)m}} \right\rbrack = \frac{\left( 1 - \frac{1}{r^{(n - 1)m}} \right)}{\left( 1 - \frac{1}{r^{m}} \right)a^{m}\left( 1 + r^{m} \right)}$

$$= \frac{\left( 1 - r^{mn - m} \right)}{\left( 1 - r^{m} \right)\left( r^{mn - m} - r^{mn - 2m} \right)}$$

✘ (a) not the correct closed form

✔ (b) telescoping gives this closed form

✔ (c) telescoping gives this closed form

✘ (d) not the correct closed form