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Question

in a sequence of 21 terms the first 11 terms are in AP with common difference 2 and the last 11 terms are in GP with common ratio 2, if the middle term of the AP is equal to the middle term of the GP , then the middle term of the entire sequence ?

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Solution

Dear Student,The first 11 term are in AP with common difference d=2 and first term is aThen the middle term of A.P. is the 6th term.i.e. a+n-1d=a+6-12=a+10Let the G.P. first term is P and common ratio r=2Then middle term of G.P.i.e. 16th term of sequence or 6th term of G.P. can be written as= prn-1=P.26-1=P.25since middle term of A.P.=middle term of G.P.Therefore a+10=P.251Also 11th term of A.P.=1st term of G.P.Therefore a+10.2=Pa+20=P2Solving 1 and 2 givesP=-1031 and a=-63031Middle term of the sequence is the 11th term of A.P. or 1st term of G.P.i.e. P=-1031Regards.

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