If {p^{th}},{q^{th}} and {r^{th}} term of an arithmetic sequence are a , b and c respectively, then

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8. Sequences and Series

easy

If the ${p^{th}},\;{q^{th}}$ and ${r^{th}}$ term of an arithmetic sequence are $a , b$ and $c$ respectively, then the value of $[a(q - r)$ + $b(r - p)$ $ + c(p - q)] = $

A

$1$

B

$- 1$

C

$0$

D

$1/2$

Solution

(c) Suppose that first term and common difference of $A.P.$'s are $A$ and $D$ respectively.

Now, ${p^{th}}$ term $ = A + (p – 1)D = a$ …..$(i)$

${q^{th}}$term $ = A + (q – 1)D = b$ ……$(ii)$

and ${r^{th}}$ term $ = A + (r – 1)D = c$ …..$(iii)$

So, $a(q – r) + b(r – p) + c(p – q)$

$ = a\left\{ {\frac{{b – c}}{D}} \right\} + b\left\{ {\frac{{c – a}}{D}} \right\} + c\left\{ {\frac{{a – b}}{D}} \right\}$

$ = \frac{1}{D}(ab – ac + bc – ab + ca – bc) = 0$.

Standard 11

Mathematics

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