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in Binomial Theorem by (4.0k points)
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Show that the coefficient of middle term in the expansion of (1 + x) 20 is equal to the sum of the coefficients of two middle terms is the expansion of (1 + x) 19

1 Answer

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Best answer

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.

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