# Miscellaneous Exercise [NCERT, pg 147]

1. If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and $\displaystyle \sum_{x=1}^{n} f(x)=120$,find the value of n.

2. The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
3. The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
4. The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

5. A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

6. If $\frac{a+bx}{a-bx}=\frac{b+cx}{b-cx}=\frac{c+dx}{c-dx}$(x≠ 0), then show that a, b, c and d are in G.P.

7. Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that $P^2R^n = S^n$.

8. If a, b, c, d are in GP, prove that $(a^n + b^n), (b^n + c^n), (c^n + d^n)$ are in GP

9. If a and b are the roots of $x^2 – 3x + p = 0$ and c, d are roots of $x^2 – 12x + q = 0$, where a, b, c, d form a GP. Prove that (q + p) : (q – p) = 17:15.

10. The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that $a:b=\left ( m+\sqrt{m^2-n^2} \right ) : \left ( m-\sqrt{m^2-n^2} \right )$

11. Find the sum of the following series up to n terms:
(i) 5 + 55 +555 + …
(ii) 0.6 +0.66 +0.666+…
12. Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + ... + n terms.

13. A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much will the tractor cost him?
14. Bidhan buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?

15. A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.