11th Class Mathematics Other Series Question Bank Critical Thinking

  • question_answer If the sum of first \[n\] terms of an A.P. be equal to the sum of its first \[m\] terms, \[(m\ne n)\], then the sum of its first \[(m+n)\] terms will be [MP PET 1984]

    A)   0

    B) \[n\]

    C) \[m\]

    D) \[m+n\]

    Correct Answer: A

    Solution :

    As given  \[\frac{n}{2}\left\{ 2a+(n-1)d \right\}=\frac{m}{2}\left\{ a+(m-1)d \right\}\] \[\Rightarrow \]\[2a(m-n)+d({{m}^{2}}-m-{{n}^{2}}+n)=0\] \[\Rightarrow \] \[(m-n)\left\{ 2a+d(m+n-1) \right\}=0\] \[\Rightarrow \] \[2a+(m+n-1)d=0\], \[(\because \ m\ne n)\] \[\therefore \] \[{{S}_{m+n}}=\frac{m+n}{2}\left\{ 2a+(m+n-1)d \right\}=\frac{m+n}{2}\left\{ 0 \right\}=0\].


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