11th Class Mathematics Other Series Question Bank Critical Thinking

  • question_answer Three numbers form a G.P. If the \[{{3}^{rd}}\] term is decreased by 64, then the three numbers thus obtained will constitute an A.P. If the second term of this A.P. is decreased by 8, a G.P. will be formed again, then the numbers will be

    A) 4, 20, 36

    B) 4, 12, 36

    C) 4, 20, 100

    D) None of the above

    Correct Answer: C

    Solution :

    \[a,\ ar,\ a{{r}^{2}}\] are in  G.P. \[a,\ ar-8,\ a{{r}^{2}}-64\] are in A.P., we get \[\Rightarrow \]\[a({{r}^{2}}-2r+1)=64\]                       .....(i) Again, \[a,\ ar-8,\ a{{r}^{2}}-64\] are in G.P. \[\therefore \]\[{{(ar-8)}^{2}}=a(a{{r}^{2}}-64)\] or  \[a(16r-64)=64\] .....(ii) Solving (i) and (ii), we get\[r=5,\ a=4\]. Thus required numbers are 4, 20, 100. Trick: Check by alternates according to conditions (a) \[\Rightarrow \]4, 20, - 28 which are not in A.P. (b) \[\Rightarrow \]4, 12, - 28 which are also not in A.P. (c) \[\Rightarrow \]4, 20, 36 which are obviously in A.P. with 16 as common difference. These numbers also satisfy the second condition \[i.e.\] 4, 20 - 8, 36 are in G.P.

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