The index 20 in (1 + x) 20 in (1 +x) 20 is even.
Middle term = \(\frac{T^{20+2}}{2}\)
= T10 + 1 = 20C10 (1)10 x10
= 20C10 x10
Now coefficient of middle term in (1 +x) 19
The index 19 in (1 +x) 19 is odd.
So, middle term as \(\frac{T^{19+1}}{2}\) and the next term i.e., T10 and T11
T10 = = T9 + 1 = 19C9 × 110 × x9 = 19C9 x9
And T11 = T10 + 1 = 19C10 × x10 = 19C10 x10
Now, sum of coefficient of middle terms in
(1 + x) 19 = 19C9 + 19C10 = 20C10
= coefficient of middle term in (1 + x) 20
= Hence Proved.