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**17. If **** are in G.P., prove that **** are in G.P.**

**Ans. **Given: are in G.P.

To prove: are in G.P.

Let

And

Also

Now,

Therefore, are in G.P.

**18. If **** and **** are the roots **** and **** are roots of **** where **** form a G.P. Prove that **** **

**Ans. **Let

And

Also

and are the roots

……….(i)

And

……….(ii)

Also are roots of

……….(iii)

And

……….(iv)

Dividing eq. (iii) by eq. (i),

Now

=

Therefore,

**19. The ratio of the A.M. and G.M. of two positive numbers **** and **** is **** Show that **** = **** **

**Ans. **Given:

By component and dividend,

Again by component and dividend,

Therefore, =

**20. If **** are in A.P.; **** are in G.P. and **** are in A.P., prove that **** are in G.P.**

**Ans. **Since, are in A.P.

Since, are in G.P.

……….(i)

Also are in A.P.

Putting values of and in eq. (i),

which shows that are in G.P.

**21. Find the sum of the following series up to **** terms:**

**(i) 5 + 55 + 555 + ……….**

**(ii) .6 + .66 + .666 + ………….**

**Ans. (i)** = 5 + 55 + 555 + ………. up to terms

= 5 [1 + 11 + 111 + ………. up to terms]

=

=

=

=

=

**(ii)** = .6 + .66 + .666 + …………. up to terms

= 6 [.1 + .11 + .111 + ………. up to terms]

=

=

=

=

=

=

=

**22. Find the 20 ^{th} term of the series **

**terms.**

**Ans. **Given: terms

=

**23. Find the sum of the first **** terms of the series: 3 + 7 + 13 + 21 + 21 + ……….**

**Ans. **Given: ……….(i)

Also ……….(ii)

Subtracting eq. (ii) from eq. (i),

=

=

=

=

=

=

=

**24. If ****are the sum of first **** natural numbers, their squares and their cubes respectively, show that **** **

**Ans. **Here , and

Now,

Proved.

**25. Find the sum of the following series up to **** terms: **** **

**Ans. **Given: up to terms

=

=

=

=

=

=

=

**26. Show that **** **

**Ans. **Given:

= =

=

=

=

= = =

**27. 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 the tractor cost him?**

**Ans. **Total cost of the tractor = Rs. 12000, Cash amount paid = Rs. 6000

Balance amount = 12000 – 6000 = Rs. 6000

Interest of 1^{st} installment = = Rs. 720

Amount of 1^{st} installment = 500 + 720 = Rs. 1220

Interest of 2^{nd} installment = = Rs. 660

Amount of 2^{nd} installment = 500 + 660 = Rs. 1160

Interest of 3^{rd} installment = = Rs. 600

Amount of 3^{rd} installment = 500 + 600 = Rs. 1100

Sequence of installments is 1220, 1160, 1100, ……….

Here, and

=

= 6 [2440 – 660] = Rs. 10680

Therefore, the total cost of tractor is (10680 + 6000) = Rs. 16680.

**28. Shams had Ali buys a scooter for Rs. 22000. He pays Rs. 4000 cash and agrees to pay the balance in annual installment of Rs. 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?**

**Ans. **Total cost of the scooter = Rs. 22000, Cash amount paid = Rs. 4000

Balance amount = 22000 – 4000 = Rs. 18000

Interest of 1^{st} installment = = Rs. 1800

Amount of 1^{st} installment = 1000 + 1800 = Rs. 2800

Interest of 2^{nd} installment = = Rs. 1700

Amount of 2^{nd} installment = 1000 + 1700 = Rs. 2700

Interest of 3^{rd} installment = = Rs. 1600

Amount of 3^{rd} installment = 1000 + 1600 = Rs. 2600

Sequence of installments is 2800, 2700, 2600, ……….

Here, and

=

= 9 [5600 – 1700] = Rs. 35100

Therefore, the total cost of tractor is (35100 + 4000) = Rs. 39100.

**29. 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.**

**Ans. **Total letters in the first set = 4, Total letters in the second set = = 16

Total letters in the third set = = 64

Sequence of letters is 4, 16, 64, ………. In G.P.

Here and

=

=

= = 87380

Since, the amount of postage on each letter is 50 paise.

Therefore, amount spent on postage = = Rs. 43690.

**30. A man deposited Rs. 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.**

**Ans. **Total amount deposited = Rs. 10000, Rate of interest = 5% per annum

Interest of first year = = Rs. 500

Here

Amount in 15^{th} year = = 10000 + 7000 = Rs. 17000

Total amount after 20 years =

= 10 (20000 + 9500) = Rs. 295000

**31. A manufacturer reckons that the value of a machine, which cost him Rs. 15625 will depreciate each year by 20%. Find the estimated value at the end of 5 years.**

**Ans. **After 1 year value of machine = = 15625 – 3125 = Rs. 12500

After 2 year value of machine = = 12500 – 2500 = Rs. 10000

After 3 year value of machine = = 10000 – 2000 = Rs. 8000

Sequence of values of machine is 12500, 10000, 8000, ……. in G.P.

Here and

= = Rs. 5120

Therefore, the value of machine at the end of 5 years is Rs. 5120.

**32. 150 workers were engaged to finish a job in a certain number of boys. 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.**

**Ans. **Number of workers on the first day = 150

Number of workers on the second day = 150 – 4 = 146

Number of workers on the third day = 146 – 4 = 142

Sequence of values of workers is 150, 146, 142, ……… in A.P.

Here

Total number of workers required to finish the work in days

=

= ……….(i)

If no worker had dropped out, then

Total number of workers required to finish the work days = ……….(ii)

From eq. (i) and (ii),

and which is not possible

Therefore, the work is completed in 25 days.