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Arithmetic Progressions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSArithmetic Progressions3 Marks
Question
If $\frac{\text{b}+\text{c}}{\text{a}},\ \frac{\text{c}+\text{a}}{\text{b}},\ \frac{\text{a}+\text{b}}{b}$ are A.P., prove that: bc, ca, ab are in A.P.
✓

Answer

$\text{bc},\ \text{ca},\ \text{ab}$ are in A.P. Then, $\text{ca}-\text{bc}=\text{ab}-\text{ca}$ $\text{c}(\text{a}-\text{b})=\text{a}(\text{b}-\text{c})\ .....(1)$ If $\frac{1}{​​​\text{a}​},\ \frac{1}{\text{b}},\ \frac{1}{\text{c}}$ are in A.P $\frac{1}{\text{b}}-\frac{1}{\text{a}}=\frac{1}{\text{c}}-\frac{1}{\text{b}}$ $\Rightarrow\text{c}(\text{a}-\text{b})=\text{a}(\text{b}-\text{c})\ .....(2)$ Thus, the condition necessare to prove $\text{bc},\ \text{ca},\ \text{ab}$ iv A.P is full filled. Thes, $\text{bc},\ \text{ca},\ \text{ab}$ are in A.P

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