Name: Sum of Infinite Terms of G.P.
Uploaded: 2019-07-19
Duration: 3 min 36 s
Description: Do you know that we can find the sum of an infinite number of terms of G.P.? Of course, there will be some conditions. Check this video to learn more.

Question

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.

Accepted Answer

Given: $a^{2}, b^{2}, c^{2}$ are in A.P.

$\Rightarrow b^{2} - a^{2} = c^{2} - b^{2}$

$\Rightarrow (b - a) (b + a) = (c - b) (c + b)$

$\Rightarrow \frac{(b - a)}{(c + b)} = \frac{(c - b)}{(b + a)}$

$\Rightarrow \frac{(b - a)}{(c + b) (c + a)} = \frac{(c - b)}{(b + a) (c + a)}$ [Using the property if each term of a given A.P is divided by non-zero term then resulting sequence is also an A.P.]

$\Rightarrow \frac{(b + c - c - a)}{(c + b) (c + a)} = \frac{(c + a - a - b)}{(b + a) (c + a)}$

$\Rightarrow \frac{1}{c + a} - \frac{1}{b + c} = \frac{1}{a + b} - \frac{1}{c + a}$

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

$\Rightarrow \frac{a + b + c}{b + c}, \frac{a + b + c}{c + a}, \frac{a + b + c}{a + b}$ are in A.P. [Using the property if each term of a given A.P is multiplied by non-zero term then resulting sequence is also an A.P.]

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

$\Rightarrow \frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}$ are in A.P. [Using the property if constant is subtracted from each term of a given A.P then the resulting sequence is also an A.P.]

Hence proved.

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.

Important Questions on Sequence and Series

Learn and Prepare for any exam you want !

If a2,b2,c2 are in A.P, show that ab+c,bc+a,ca+b are in A.P.

Important Questions on Sequence and Series

Learn and Prepare for any exam you want !

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.