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

 Exercise 17.4

Page no 17.27

Question 1

If $\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{a+b}$ 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

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

Question 3

If $a, b, c$ 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

If $\frac{b+c-a}{a}, \frac{c+a-b}{b}, \frac{a+b-c}{c}$ 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

If $(b-c)^{2},(c-a)^{2},(a-b)^{2}$ 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

If $a, b, c$ 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.