Find a, b and n in the expansion of (a + b)ⁿ if the first three terms of the expansion are 729, 7290 and 30375, respectively

Solution:

Using the binomial theorem, the first three terms in the expansion of (a + b)ⁿ are 

ⁿC₀ aⁿ = aⁿ = 729.. (1)

ⁿC₁ a^{n - 1} b =  n a^{n - 1} b = 7290 ... (2)

ⁿC₂ a^{n - 2} b² =  n(n - 1)/2 (a^{n - 2} b²) = 30375 ... (3)

We divide (2) by (1), then we get

(n a^{n - 1} b) / aⁿ = 7290 / 729

nb / a = 10

b / a = 10/n ... (4)

Now, we divide (3) by (2), then we get

[n(n - 1)/2 (a^{n - 2} b²)] / [n a^{n - 1} b] = 30375 / 7290

(n-1)/2 (b / a) = 25/6 

(n-1)/2 (10/n) = 25 /6 (from (4))

60(n - 1) = 50 n

60n - 60 = 50n

10n = 60

n = 6

Substitute it in (1),

a⁶ = 729 = 3⁶ ⇒ a = 3

Substitute n = 6 and a = 4 in (4),

b / 3 = 10 / 6

b = 30/6 = 5

Therefore, n = 6, a = 3 and b = 5

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NCERT Solutions Class 11 Maths Chapter 8 Exercise ME Question 1

Find a, b and n in the expansion of (a + b)ⁿ if the first three terms of the expansion are 729, 7290 and 30375, respectively.

Summary:

a, b and n in the expansion of (a + b)ⁿ if the first three terms of the expansion are 729, 7290 and 30375, respectively are 3, 5, and 6 respectively.