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Show that the sum of (m + n) th and (m – n) th terms of an A.P. is equal to twice the m th term.

1.  Show that the sum of ( m+n)^{th} and ( m-n)^{th} terms of an A.P. is equal to twice the m^{th}term.

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Let a be first term and d be common difference of AP.

Kth term of a AP is given by,

a_k=a+(k-1)d

\therefore a_m_+_n=a+(m+n-1)d

\therefore a_m_-_n=a+(m-n-1)d

a_m=a+(m-1)d

a_m_+_n+ a_m_-_n=a+(m+n-1)d+a+(m-n-1)d

                           =2a+(m+n-1+m-n-1)d

                         =2a+(2m-2)d

                       =2(a+(m-1)d)

                        =2.a_m

Hence, the sum of ( m+n)^{th} and ( m-n)^{th} terms of an A.P. is equal to twice the m^{th}term.

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