KC Sinha Mathematics Solution Class 11 Chapter 17 Sequences and Series : AP Exercise 17.4

Exercise 17.4

Page no 17.27

Question 1

$\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}$ are in A.P. and

$a+b+c \neq 0$ , prove that

$\frac{1}{b+c}, \frac{1}{c+a}, \frac{1}{a+h}$ are in A.P.

Question 2

$a^{2}, b^{2}, c^{2}$ are in A.P., show that

$\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}$ are in A.P.

Question 3

$a, b, c$ are in A.P. prove that :

(i)

$\frac{1}{b c}, \frac{1}{c a}, \frac{1}{a b}$ are in A.P.

(ii)

$(b+c)^{2}-a^{2},(c+a)^{2}-b^{2}+(a+b)^{2}-c^{2}$ are in A.P.

(iii)

$\frac{1}{\sqrt{b}+\sqrt{c}}, \frac{1}{\sqrt{c}+\sqrt{a}}, \frac{1}{\sqrt{a}+\sqrt{b}}$ are in A.P.

Question 4

$\frac{b+c-a}{a}, \frac{c+a-b}{b}, \frac{a+b-c}{c}$ are in A.P., show that

$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P. provided

$a+b+c \neq 0$ .

Question 5

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

$\frac{1}{b-c}, \frac{1}{c-a}, \frac{1}{a-b}$ are in A.P.

Question 6

$a, b, c$ are in A.P. show that

(i)

$(a-c)^{2}=4(a-b)(b-c)$

(ii)

$a^{3}+c^{3}+6 a b c=8 b^{3}$

(iii)

$(a+2 b-c)(2 b+c-a)(c+a-b)=4 a b c$ .

Hints to selected problems :

5. Add

$a b+b c+c a-a^{2}-b^{2}-c^{2}$ to each term.

alternatively, let

$\alpha=b-c, \beta=c-a, \gamma=a-b$ , then

$\alpha+\beta+\gamma=0$

6. Put

$b=\frac{a+c}{2}$ on L.H.S. and R.H.S. in all the parts.