Question

1. The product of first 3 terms of a GP is 1000. If 6 terms to the second term and 7 is added to the third term, the terms become an AP.

(a) Find the second term of GP. (1)

(b) Find the terms of the GP. (2)

2. Find the sum of n terms of the series 7 + 77 + 777 +……   … (3)

Answer 1

1. 

(a) \(\frac{a}{r}\)× a × ar = 1000 ⇒ a = 10

(b) \(\frac{a}{r}\), a + 6, ar + 7 from an AP

(a + 6) – \(\frac{a}{r}\) = (ar + 7) – (a + 6)

16 – \(\frac{10}{r}\) = 10r + 7 – 16

⇒ 16r – 10 = 10r² – 9r

⇒ 10r² – 25r + 10 = 0 ⇒ r = 2; \(\frac{1}{2}\)

Hence, 20, 10, 5 and 5, 10, 20;

2. S_n = 7 + 77 + 777 +……………

= 7(1 + 11 + 111 +…………)

= \(\frac{7}{9}\)(9 + 99 + 999 +………..)

= \(\frac{7}{9}\)(10 – 1 + 100 – 1 + 1000 – 1 +………..)

= \(\frac{7}{9}\)(10 + 100 + 1000+……….-1 – 1 – 1-………)