Multiple Choice Question (MCQ)

If \{a_n\} is an arithmetic sequence such that S_m=S_n, then the value of S_{m+n} is

  1. ×

    S_m+S_n

  2. ×

    \dfrac{1}{2}(S_m+S_n)

  3. ×

    \sqrt{S_mS_n}

  4. 0

Collected in the board: Arithmetic sequence

Steven Zheng posted 1 month ago


Answer

  1. S_{m+n}=\dfrac{(S_m-S_n)(m+n)}{m-n}


    S_{m}-S_{n} = a_1m+\dfrac{m(m-1)d}{2}-a_1n-\dfrac{n(n-1)d}{2}

    =a_1(m-n)+\dfrac{(m^2-n^2-m+n)d}{2}

    \dfrac{S_{m}-S_{n}}{m-n} = a_1+\dfrac{(m+n-1)d}{2}

    \dfrac{(S_{m}-S_{n})(m+n)}{m-n}

    =a_1(m+n)+\dfrac{(m+n)(m+n-1)d}{2}

    = S_{m+n}

    Therefore,

    S_{m+n}=\dfrac{(S_m-S_n)(m+n)}{m-n}

    If S_{m}=S_{n}, S_{m+n} = 0

Steven Zheng posted 1 month ago

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