Question

Given that the coefficient of the third term in the expansion of (1+3x)^n, in ascending powers of x, is 1539, find the value of n where n is a positive integer ​

Answer 1

n = 19

Explanation:

`(1 + 3x)^(n) = {1}^(n) + {1}^(n - 1) .(3x)^(1).nC1 + {1}^(n - 2).(3x)^(2) .nC2 + ...`

Third term coefficient:

`1^(n - 2).(3)^(2) .nC2 = 1539 \\ 9.nC2 = 1539 \\ nC2 = 171 \\`

nCr = n!/((n - r)!r!)

nC2 = 171

n!/((n - 2)!2!) = 171

n!/2(n - 2)! = 171

n!/(n - 2)! = 342

n.(n - 1).(n -2).../((n - 2).(n - 3).(n - 4)...) = 342

n(n - 1) = 342

n² - n = 342

(n - ½)² - ¼ = 342

(n - ½)² = 1369/4

(n - ½) = ±37/2

n = 1/2 ± 37/2

n = -18 or n = 19

Since we are told n is a positive integer, it is 19