if number of terms in an a p is odd then prove that middle term of a p - Maths - Sequences and Series

Question

if number of terms in an a p is odd then prove that middle term of
a p is a m of first term and last term of that a p

Vijay Kumar Gupta · Accepted Answer

Dear student,

Please find below the solution ot the asked query:

Let the first term and common difference of an AP be 'a' and 'd' respectively
Since the number of terms are odd, the nth term is 2n+1So the last term of the AP is given by,   an=a+2n+1-1d=a+2ndThe arithmetic mean of first and last term is,   AM=a1+an2       =a+a+2nd2       =2a+2nd2       =2a+nd2       =a+nd         
1The middle term if there are 2n+1 terms in AP is,    2n+1+12th term=n+1th termFind n+1th term of the AP   an+1=a+n+1-1d  an+1=a+nd        
2From 1  and 2, it can be concluded that middle term of an AP consisiting of odd no of terms  is the AM of the first and last term

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if number of terms in an a.p. is odd then prove that middle term of a.p. is a.m. of first term and last term of that a.p.