Question

Show that the progression $$-27, 9, -3, 1, \frac{-1}{3}$$, ......... is a G.P.

Write its (i) first term (ii) common ratio (iii) nth term (iv) 9th term.

Answer 1

To show it is a G.P.

$$\frac{9}{-27} = -\frac{1}{3}, \frac{-3}{9} = -\frac{1}{3}, \frac{1}{-3} = -\frac{1}{3}, \frac{\frac{-1}{3}}{1} = -\frac{1}{3}$$

All the ratios are the same (constant = $$-\frac{1}{3}$$), so the given numbers form a G.P.

(i) First term: $$a = -27$$

(ii) Common ratio: $$r = -\frac{1}{3}$$

(iii) nth term: $$T_n = ar^{n - 1} = -27 \times \left(-\frac{1}{3}\right)^{n - 1}$$

(iv) 9th term: $$T_9 = -27 \times \left(-\frac{1}{3}\right)^{8}$$

$$= -27 \times \frac{1}{3^8}$$

$$= -\frac{27}{6561}$$

$$= -\frac{1}{243}$$

So, the 9th term is $$-\frac{1}{243}$$.