In a sequence of `(4n+1)` terms, the first `(2n+1)` terms are n 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 terms of the A.P. and LG.P. are equal ,then the middle terms of the sequence is `(n .2 n+1)/(2^(2n)-1)` b. `(n .2 n+1)/(2^n-1)` c. `n .2^n` d. none of these

Explanation

To solve the problem, we need to find the middle term of a sequence consisting of \(4n + 1\) terms, where the first \(2n + 1\) terms are in an arithmetic progression (A.P.) with a common difference of 2, and the last \(2n + 1\) terms are in a geometric progression (G.P.) with a common ratio of 0.5. We are given that the middle terms of the A.P. and G.P. are equal.

### Step-by-Step Solution:

1. **Identify the Middle Term of the A.P.**:
- The first \(2n + 1\) terms of the sequence are in A.P. The middle term of an A.P. with \(2n + 1\) terms is the \((n + 1)\)-th term.
- The \(k\)-th term of an A.P. can be expressed as:
\[
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