Question

Show that the progression 2, 6, 18, 54, 162, ......... is a G.P.

Write its (i) first term (ii) common ratio (iii) nth term (iv) 8th term.

Answer 1

To show it is a G.P.

Divide each term by the term before it:

$$\frac{6}{2} = 3, \frac{18}{6} = 3, \frac{54}{18} = 3, \frac{162}{54} = 3$$

All the ratios are the same (constant = 3), so the given numbers form a G.P.

(i) First term: $$a = 2$$

(ii) Common ratio: $$r = \frac{6}{2} = 3$$

(iii) nth term: $$T_n = ar^{n - 1} \Rightarrow T_n = 2 \times 3^{n - 1}$$

(iv) 8th term: $$T_8 = 2 \times 3^{(8 - 1)}$$ 

$$= 2 \times 3^7$$ 

$$= 2 \times 2187$$ 

$$= 4374$$

So, the 8th term is 4374.