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JEE Main & Advanced Mathematics Sequence & Series Question Bank Arithmetic Progression

  • question_answer
    If the \[{{p}^{th}},\ {{q}^{th}}\] and \[{{r}^{th}}\] term of an arithmetic sequence are a , b and \[c\] respectively, then the value of \[[a(q-r)\] + \[b(r-p)\] \[+c(p-q)]=\] [MP PET 1985]

    A) 1

    B) \[-1\]

    C) 0

    D) 1/2

    Correct Answer: C

    Solution :

    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, \[a(q-r)+b(r-p)+c(p-q)\] \[=a\left\{ \frac{b-c}{D} \right\}+b\left\{ \frac{c-a}{D} \right\}+c\left\{ \frac{a-b}{D} \right\}\] \[=\frac{1}{D}(ab-ac+bc-ab+ca-bc)=0\].


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