Question

Show that the progression $$2, 2\sqrt{2}, 4, 4\sqrt{2}$$, ......... is a G.P.

Write its (i) first term (ii) common ratio (iii) nth term (iv) 11th term.

Answer 1

To show it is a G.P.

$$\frac{2\sqrt{2}}{2} = \sqrt{2}, \frac{4}{2\sqrt{2}} = \frac{4}{2 \times 1.414...} = \sqrt{2}, \frac{4\sqrt{2}}{4} = \sqrt{2}$$

(Check: $$2\sqrt{2} \times \sqrt{2} = 2 \times 2 = 4$$, so the middle ratio is also $$\sqrt{2}$$.)

All the ratios are the same (constant = $$\sqrt2$$), so the given numbers form a G.P.

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

(ii) Common ratio: $$r = \sqrt{2}$$

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

(iv) 11th term: $$T_{11} = 2 \times (\sqrt{2})^{10}$$

$$= 2 \times \left((\sqrt{2})^2\right)^5$$

$$= 2 \times 2^5$$

$$= 2 \times 32$$

$$= 64$$

So, the 11th term is 64.