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Arithmetic Progressions — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSArithmetic Progressions3 Marks
Question
If $\text{a}^2,\ \text{b}^2,\ \text{c}^2$ are in A.P., prove that $\frac{\text{a}}{\text{b}+\text{c}},\frac{\text{b}}{\text{c}+\text{a}},\frac{\text{c}}{\text{a}+\text{b}}$ are in A.P.
✓

Answer

$\frac{\text{a}}{\text{b}+\text{c}},\frac{\text{b}}{\text{c}+\text{a}},\frac{\text{c}}{\text{a}+\text{b}}$ are in A.P if $\frac{​​\text{b}}{​​\text{a}+​​\text{c}}-\frac{​​\text{a}}{​​\text{b}+​​\text{c}}=\frac{​​\text{c}}{​​\text{a}+​​\text{b}}-\frac{​​\text{b}}{​​\text{a}+​​\text{c}}$
$​​\text{LHS}=\frac{​​\text{b}}{​​\text{a}+​​\text{c}}-\frac{​​\text{a}}{​​\text{b}+​​\text{c}}$
$\Rightarrow\frac{\text{b}^2+\text{bc}-\text{a}^2-\text{ac}}{(\text{a}+\text{c})(\text{b}+\text{c})}$
$\Rightarrow\frac{(\text{b}-\text{a})(\text{a}+\text{b}+\text{c})}{(\text{a}+\text{c})(\text{b}+\text{c})}\ .....(1)$
$\text{RHS}=\frac{\text{a}}{\text{a}+\text{b}}=\frac{\text{b}}{\text{a}+\text{c}}$
$\Rightarrow\frac{\text{ca}+\text{c}^2-\text{b}^2-\text{ab}}{(\text{a}+\text{b})(\text{b}+\text{c})}$
$\Rightarrow\frac{(\text{c}-\text{d})(\text{a}+\text{b}+\text{c})}{(\text{a}+\text{b})(\text{b}+\text{c})}\ .....(2)$
and
$\text{a}^2,\ \text{b}^2,\ \text{c}^2$ are in A.P
$\therefore\text{b}^2-\text{a}^2=\text{c}^2-\text{b}^2\ .....(3)$
Substituting $\text{b}^2-\text{a}^2$ with $\text{c}^2-\text{b}^2$
$(1)=(2)$
$\therefore\frac{\text{a}}{\text{b}+\text{c}},\ \frac{\text{b}}{\text{a}+\text{c}},\ \frac{\text{c}}{\text{a}+\text{b}}$ are in A.P

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