Q110 / 200 · WBJEE 2013 · Mathematics

Five numbers are in AP with common difference $\neq 0 .$ If the 1 st, 3 rd and 4 th terms are in GP, then

  • Correct answer
    A
    the 5th term is always $0 .$
  • B
    the 1 st term is always $0 .$
  • C
    the middle term is always $0 .$
  • D
    the middle term is always -2.

Solution

Let the tine number is in AP are $$ (a-2 d),(a-d), a,(a+d),(a+2 d) $$ where, $d \neq 0$ Given, 1 st, $3 r d$ and 4 th terms are in GP. $$ \begin{array}{lr} \Rightarrow \quad a^{2}=(a-2 d)(a+d) \\ \Rightarrow \quad a^{2}=a^{2}-2 a d+a d-2 d^{2} \\ \Rightarrow \quad 2 d^{2}+a d=0 \Rightarrow d(2 d+a)=0 \\ \because \quad d \neq 0 \end{array} $$ $\therefore \quad a+2 d=0 \Rightarrow a=-2 d$ Hence, terms are $$ -4 d,-3 d,-2 d,-d, 0 $$ $\therefore$ The fifth term is always $0 .$

Hence, the correct answer is option: 'A'.

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