If the p^{th},q^{th},and r^{th} terms of an a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the first term be $A$ and the common difference be $D$ for the arithmetic progression.The $p^{th}$ term is $A + (p - 1)D = a$ $(i)$The $q^{th}

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(C) Let the first term be $A$ and the common difference be $D$ for the arithmetic progression.The $p^{th}$ term is $A + (p - 1)D = a$ $(i)$The $q^{th}$ term is $A + (q - 1)D = b$ $(ii)$The $r^{th}$ term is $A + (r - 1)D = c$ $(iii)$Subtracting $(ii)$ from $(i)$,$(i)$ from $(iii)$,and $(iii)$ from $(ii)$ gives:$a - b = (p - q)D$$b - c = (q - r)D$$c - a = (r - p)D$Now,consider the expression $E = a(q - r) + b(r - p) + c(p - q)$.Substituting the values of $(q - r)$,$(r - p)$,and $(p - q)$ from the differences above:$E = a\left(\frac{b - c}{D}\right) + b\left(\frac{c - a}{D}\right) + c\left(\frac{a - b}{D}\right)$$E = \frac{1}{D} [a(b - c) + b(c - a) + c(a - b)]$$E = \frac{1}{D} [ab - ac + bc - ab + ca - bc]$$E = \frac{1}{D} [0] = 0$.

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

Similar Questions

If $a_1, a_2, a_3, \dots$ are in $A.P.$ such that $a_1 + a_7 + a_{16} = 40$,then the sum of the first $15$ terms of this $A.P.$ is

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an Arithmetic Progression are $a$,$b$,and $c$ respectively,then $[a(q - r) + b(r - p) + c(p - q)] = ?$

If the $n^{th}$ term of an arithmetic progression is $\frac{(2n + 1)}{3}$,what is the sum of its first $19$ terms?

If the sum of $n$ terms of an $A.P.$ is $3n^{2} + 5n$ and its $m^{\text{th}}$ term is $164$,find the value of $m$.

If $n$ arithmetic means are inserted between $a$ and $100$ such that the ratio of the first mean to the last mean is $1:7$ and $a+n=33$,then the value of $n$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module