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Arithmetic Progressions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSArithmetic Progressions3 Marks
Question
If $\frac{1}{\text{a}},\ \frac{1}{\text{b}},\ \frac{1}{\text{c}}$ are in A.P., Prove that: $​\text{a}(​\text{b}+​\text{c}),\ ​\text{b}(​\text{c}+​\text{a}),\ ​\text{c}(​\text{a}+​\text{b})$ are in A.P.
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Answer

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

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