Question

if the sum of first m terms of an ap vanishes show the sum of next ap is -an( m+n) /m+1 where a is first term
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Answer 1

Answer:

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Step-by-step explanation:

Let a be the first term and d be the common difference of the given AP. Then,

     Sm=m/2[2a+(m−1)d]      

       0 =m/2[2a+(m-1)d]        ---------------------(1)

       0=2a+(m-1)d

      -2a=(m-1)d

     -2a/m-1=d           ----------------------------(2){∵m≠0}

     Also,

    sum of next n terms=sum of the first (m + n)-sum of first terms

                                          Sn=s(m + n)-Sm

               Sn=(m + n)/2[2a+(m+n−1)d]-0            --------- [from (1) ]

               Sn=(m + n)/2[2a+(m+n-1) -2a/m-1]     --------- [ from (2) ]

              Sn=2a(m + n)/2  [1-(m+n-1/m-1)]

              Sn=a(m + n)/m-1  [m-1-m-n+1]

              Sn=a(m + n)/m-1 [-n]

               Sn=-an( m + n) /m+1  

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