Question

In the binomial expression of (1+x)

n
the first three terms are 1+3+4+---. Calculate
the numerical values of n and x, and the values of the fourth term of the expression.

Answer 1

• x = 1/3
• n = 9
• fourth term = 28/9

Explanation:

Given the first three terms of the expansion of (1 +x)^n are 1 +3 +4, you want the values of x and n, and the next term.

Binomial expansion

The first few terms of the binomial expansion of (1 +x)^n are ...

`(1+x)^n=1^n+n\cdot1^(n-1)x+(n(n-1))/(2)1^(n-2)x^2+(n(n-1)(n-2))/(2\cdot3)1^(n-2)x^3+\dots`

Comparing terms

Comparing the terms to those given, we have ...

`nx=3\\\\(nx)((n-1)x)/2=4`

Expanding the second of these equations, and substituting the first, we get ...

`(nx)(nx -x)=8\qquad\text{multiply by 2}\\\\3(3-x)=8\qquad\text{substitute $nx=3$}\\\\9-8=3x\qquad\text{add $3x-8$}\\\\\boxed{x=(1)/(3)}\qquad\text{divide by 3}\\\\(1)/(3)n=3\qquad\text{substitute for $x$ in $nx=3$}\\\\\boxed{n=9}\qquad\text{multiply by 3}`

Fourth term

Then the fourth term is ...

`(n(n-1)(n-2))/(6)x^3=(9\cdot8\cdot7)/(6)\cdot\left((1)/(3)\right)^3=\boxed{(28)/(9)}`

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Additional comment

Then the expansion is ...

(1 +1/3)^9 = 1 + 3 + 4 + 28/9 + 14/9 + 14/27 + ...

The n-th term is (11-n)/(3(n-1)) times the term before.