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The MIDDLE TERM in the expansion of $(a+b)^{2n}$ (where the power is EVEN) is
A$T_n$
B$T_{2n}$
C$T_{n+2}$
D$T_{n+1}$
Answer & Solution
Correct answer: D. $T_{n+1}$
1. The expansion of $(a+b)^N$ has $N+1$ terms.
2. When $N$ is EVEN (here $N = 2n$), there is ONE middle term — namely the $(N/2 + 1)$-th term.
3. With $N = 2n$: middle term is $T_{2n/2 + 1} = T_{n+1}$.
4. Example: $(a+b)^4$ has $5$ terms; middle is $T_3$ (since $n=2$, $T_{n+1} = T_3$). Verify: $T_3 = \binom{4}{2}\,a^2 b^2 = 6 a^2 b^2$ ✓.
5. Option A is one term before. Option C is one term after. Option D is the last term.
6. Note: when $N$ is ODD, there are TWO middle terms — $T_{(N+1)/2}$ and $T_{(N+3)/2}$.
_Source: NCERT Class 11 Mathematics, Ch 7, §7.2.2 (Middle term), p. 6._
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