Miscellaneous Exercise on Chapter 9

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

Show that $\frac{1\times 2^2+2\times 3^2+\dots +n\times {\left(n+1\right)}^2}{1^2\times 2+2^2\times 3+\dots +n^2\times \left(n+1\right)}=\frac{3n+5}{3n+1}$

A farmer buys a used tractor for Rs$12000$ . He pays Rs $6000$ cash and agrees to pay the balance in annual installments of Rs $500$ plus $12\%$ interest on the unpaid amount. How much will be the tractor cost him?

Shamshad Ali buys a scooter for `₹ $22000$`. He pays `₹ $4000$` cash and agrees to pay the balance in an annual installment of `₹ $1000$` plus $10\%$ interest on the unpaid amount. How much will the scooter cost him?

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 $8^{th}$ set of letter is mailed.

A man deposited ₹$10000$  in a bank at the rate of $5\%$ simple interest annually. Find the amount in ${15}^{th}$ year since he deposited the amount and also calculate the total amount after  $20$ years.

A manufacturer reckons that the value of a machine, which costs him $\mathrm{₹}15625$, will depreciate each year by $20\%$. Find the estimated value at the end of $5$ years.

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.