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Arithmetic Progressions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSArithmetic Progressions4 Marks
Question
If $\text{a},\ \text{b},\ \text{c}$ are in A.P., then show that: $\text{bc}-\text{a}^2,\ \text{ca}-\text{b}^2,\ \text{ab}-\text{c}^2$ are in A.P.
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Answer

To prove $\text{bc}-\text{a}^2,\ \text{ca}-\text{b}^2,\ \text{ab}-\text{c}^2$ are in A.P $(\text{ca}-\text{b}^2)-(\text{bc}-\text{a}^2)=(\text{ab}-\text{c}^2)-(\text{ca}-\text{b}^2)$ $\text{LHS}=(\text{a}-\text{b}^2-\text{ca}+\text{a}^2)$ $=(\text{a}-\text{b})[\text{a}+\text{b}+\text{c}]\ ......(1)$ $\text{RHS}=\text{ab}-\text{c}^2-\text{ca}+\text{b}^2$ $=(\text{b}-\text{c})[\text{a}+\text{b}+\text{c}]\ .....(2)$ and since a, b, c are in ab $\text{b}-\text{c}=\text{a}-\text{b}$ $\therefore\text{LHS}=\text{RHS}$ and Thus, $\text{bc}-\text{a}^2,\ \text{ca}-\text{b}^2,\ \text{ab}-\text{c}^2$ are in A.P

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