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

Answer:

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

Step-by-step 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 ...

  $(1+x)^n=1^n+n\cdot1^{n-1}x+\dfrac{n(n-1)}{2}1^{n-2}x^2+\dfrac{n(n-1)(n-2)}{2\cdot3}1^{n-2}x^3+\dots$

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 ...

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

Fourth term

Then the fourth term is ...

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

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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.