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Arithmetic Progressions — MATHS STD 11 Science — Question

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

T.P $\text{b}+\text{c}-\text{a},\ \text{c}+\text{a}-\text{b},\ \text{a}+\text{b}-\text{c}$ are in A.P. $\text{b}+\text{c}-\text{c},\ \text{c}+\text{a}-\text{b},\ \text{a}+\text{b}-\text{c}$ are in A.P only if $(\text{c}+\text{a}-\text{b})-(\text{b}+\text{c}-\text{a})=(\text{a}+\text{b}-\text{c})-(\text{c}+\text{a}-\text{b})$ $\text{LHS}=(\text{c}+\text{a}-\text{b})-(\text{b}+\text{c}-\text{a})$ $\Rightarrow2\text{a}-2\text{b}\ .....(1)$ $\text{RHS}=(\text{a}+\text{b}-\text{c})-(\text{c}+\text{a}-\text{b})$ $\Rightarrow2\text{b}-\text{2c}\ .....(2)$ since, $\text{a},\ \text{b},\ \text{c}$ are in A.P $\therefore\text{b}-\text{a}=\text{c}-\text{b}$ or $\text{a}-\text{b}=\text{b}-\text{c}\ .....(3)$ From (1), (2) and (3) LHS = RHS Thus, given numbers $\text{b}+\text{c}-\text{a},\ \text{c}+\text{a}-\text{b},\ \text{a}+\text{b}-\text{c}$ are in A.P

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