KC Sinha Mathematics Solution Class 11 Chapter 18 Geometric Progressions Exercise 18.3

 Exercise 18.3

Page no 18.21

Type 1

Question 1

If $a, b, c, d$ are in G.., then show that

(i) $(a+b)^{2},(b+c)^{2},(c+d)^{2}$ are in G.P.

(ii) $(a-b)^{2},(b-c)^{2},(c-d)^{2}$ are in GP.

(iii) $a^{2}+b^{2}+c^{2}, a b+b c+c d, b^{2}+c^{2}+d^{2}$ are in G.P.

(iv) $\frac{1}{a^{2}+b^{2}}, \frac{1}{b^{2}+c^{2}}, \frac{1}{c^{2}+d^{2}}$ are in G.P.

(v) $a^{n}+b^{n}, b^{n}+c^{n}, c^{n}+d^{n}$ are in G.P.

Question 2

If $a, b, c$ are in G.P., then show that $a(b-c)^{2}=c(a-b)^{2}$

Question 3

If $a, b, c$ are in $\mathrm{GP}$, then show that

(i) $\left(a^{2}-b^{2}\right)\left(b^{2}+c^{2}\right)=\left(b^{2}-c^{2}\right)\left(a^{2}+b^{2}\right)$

(ii) $\log a, \log b, \log c$ are in A.P.

(iii) $a\left(b^{2}+c^{2}\right)=c\left(a^{2}+b^{2}\right)$

Type 2

Question 4

For what values of $x$, the numbers $-\frac{2}{7}, x,-\frac{7}{2}$ are in G.P.?

Question 5

Show that the products of the corresponding terms of the sequences, $a, a r, a r^{2}, \ldots, a r^{n-1}$ and $A, A R, A R^{2}, \ldots, A R^{n-1}$ form a G.P. and find its common ratio.

Question 6

A person has 2 parents, 4 grand parents, 8 great-grand parents and so on. Find the number of his ancestors during the ten generations preceding his own.

Question 7

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2 nd hour, 4 th hour and $n$th hour.

Question 8

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.

Question 9

Ali 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 the scooter will cost him ?

Question 10

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 ?