Expand Using the Binomial Theorem (1+1/x)^5

Problem

(1+1/x)5

Solution

1. Identify the components of the binomial (a+b)n where a=1 b=1/x and n=5

2. Apply the Binomial Theorem formula, which states (a+b)n=(∑_k=0^n)((n/k))*a(n−k)*bk)

3. Write out the individual terms using the combinations (5/k)) for k=0,1,2,3,4,5

(5/0)(1)5*(1/x)0+(5/1)(1)4*(1/x)1+(5/2)(1)3*(1/x)2+(5/3)(1)2*(1/x)3+(5/4)(1)1*(1/x)4+(5/5)(1)0*(1/x)5))))))

4. Calculate the binomial coefficients (5/k)) which are 1, 5, 10, 10, 5, 1$.

1*(1)*(1)+5*(1)*(1/x)+10*(1)*(1/(x2))+10*(1)*(1/(x3))+5*(1)*(1/(x4))+1*(1)*(1/(x5))

5. Simplify each term to find the final expanded form.

1+5/x+10/(x2)+10/(x3)+5/(x4)+1/(x5)

Final Answer

(1+1/x)5=1+5/x+10/(x2)+10/(x3)+5/(x4)+1/(x5)

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