If a, b, c are in G.P., prove that: (i) a(b2+c2)=c(a2+b2) (ii) a2b2c2(

1.	If a, b, c are in G.P., prove that: (i) a(b2+c2)=c(a2+b2) (ii) a2b2c2(1a3+1b3+1c3)=a3+b3+c3 (iii) (a+b+c)2a2+b2+c2=a+b+ca−b+c (iv) 1a2−b2+1b2=1b2−c2 (v) (a+2b+2c)(a−2b+2c)=a2+4c2. (v) (a+2b+2c)(a−2b+2c)=a2+4c2.
Answer» If a, b, c are in G.P., prove that: (i) $a (b^{2} + c^{2}) = c (a^{2} + b^{2})$ (ii) $a^{2} b^{2} c^{2} (\frac{1}{a^{3}} + \frac{1}{b^{3}} + \frac{1}{c^{3}}) = a^{3} + b^{3} + c^{3}$ (iii) $\frac{(a + b + c)^{2}}{a^{2} + b^{2} + c^{2}} = \frac{a + b + c}{a - b + c}$ (iv) $\frac{1}{a^{2} - b^{2}} + \frac{1}{b^{2}} = \frac{1}{b^{2} - c^{2}}$ (v) $(a + 2 b + 2 c) (a - 2 b + 2 c) = a^{2} + 4 c^{2}$ . (v) $(a + 2 b + 2 c) (a - 2 b + 2 c) = a^{2} + 4 c^{2}$ .

If a, b, c are in G.P., prove that: (i) a(b2+c2)=c(a2+b2) (ii) a2b2c2(1a3+1b3+1c3)=a3+b3+c3 (iii) (a+b+c)2a2+b2+c2=a+b+ca−b+c (iv) 1a2−b2+1b2=1b2−c2 (v) (a+2b+2c)(a−2b+2c)=a2+4c2. (v) (a+2b+2c)(a−2b+2c)=a2+4c2.

Answer»

If a, b, c are in G.P., prove that:

(i) $a (b^{2} + c^{2}) = c (a^{2} + b^{2})$

(ii) $a^{2} b^{2} c^{2} (\frac{1}{a^{3}} + \frac{1}{b^{3}} + \frac{1}{c^{3}}) = a^{3} + b^{3} + c^{3}$

(iii) $\frac{(a + b + c)^{2}}{a^{2} + b^{2} + c^{2}} = \frac{a + b + c}{a - b + c}$

(iv) $\frac{1}{a^{2} - b^{2}} + \frac{1}{b^{2}} = \frac{1}{b^{2} - c^{2}}$

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