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 AP be equal to the middle term of the GP, then the middle term of the entire sequence is
A. - 10 31
Since, the first 11 terms are in AP, d=2 a_ 11 =a+10 d=a+20 The middle term of AP is T_6=a+5 d=a+10 For the next 11 terms in GP r=2 The middle term of GP is b(2)^5 where, b is the first term of a GP which is the last term of AP b(2)^5=(a+20) 32 According to the given condition, aligned a+10 & =(a+20) 32 \\ 31 a & =10-640 \\ a & =- 630 31 aligned Middle term of entire sequence is 11th term aligned T_ 11 & = -630 31 +10 d \\ & = -630 31 +10 2= -10 31 aligned
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