Find the constant term (term independent of x) in the expansion of (2x2-5x)9 - Mathematics and Statistics

Sum

Find the constant term (term independent of x) in the expansion of (2x^2 - 5/x)^9

Solution

Let tr+1 be the constant term in the expansion of (2x^2 - 5/x)^9

We know that, in the expansion of (a+ b)n,

tr+1 = nCr an–r br

Here a = 2x2, b = -5/x, n = 9

∴ tr+1 = ""^9"C"_"r" (2x^2)^(9 - "r") ((-5)/x)^"r"

= ""^9"C"_"r" 2^(9 - "r")*x^(18 - 2"r")*(-5)^"r"*x^(-"r")

= ""^9"C"_"r" 2^(9 - "r")*(-5)^"r"*x^(18 - 3"r")

But tr+1  is a constant term

∴ power of x = 0

∴ 18 – 3r = 0

∴ r = 6

∴ the constant term

= 9C6 29–6 · (– 5)6

= 9C3 · 23 · (– 5)    ...[∵ nCr = nCn–r]

= (9 xx 8 xx 7)/(1 xx 2 xx 3) xx 8 xx 15625

= 10500000

Concept: General Term in Expansion of (a + b)n
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Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 4 Methods of Induction and Binomial Theorem
Exercise 4.3 | Q 3. (v) | Page 80