Question

In a sequence of (4n + 1) terms, the first (2n+1) terms are in A.P., whose common difference is 2 and the last 2n + 1 terms are in G.P., whose common ratio is 0.5. If the middle term of an A.P. and G.P. are equal then the middle term of the sequence is –

• A.
• B.
• C. n 2ⁿ
• D. None of these

Answer 1

Answer: (A)

Answer 2

First (2n + 1) terms of an A.P.

a, a + 2, a + 4 …. a + 2.2n

The middle term of the sequence 4n + 1 terms is (2n + 1)^th term i.e. (a + 4n)

The middle term of the A.P. of 2n + 1 terms is (n + 1)^th term i.e. a + 2n

Again the last 2n + 1 terms the first term will be (2n + 1)^th term of the A.P. i.e. a + 4n

\ G.P. is (a + 4n), (a + 4n) (0.5)_, (a + 4n) (0.5)² … (a + 4n) (0.5)²ⁿ

Middle term is (a + 4n) (0.5)ⁿ

According to given conditions

a + 2n = (a + 4n) (0.5)ⁿ

a =

Required middle term is a + 4n

Ž + 4n

Ž Ž $\frac { \mathrm { n } 2 ^ { \mathrm { n } + 1 } } { 2 ^ { \mathrm { n } } - 1 }$