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Arithmetic Progressions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsArithmetic Progressions5 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.
✓

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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