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Chapter 8: Problem 61

Describe the pattern in the exponents on \(b\) in the expansion of \((a+b)^{n}\).

Short Answer

Expert verified
In the expansion of \((a+b)^{n}\), the exponents on \(b\) start at 0 and increase by 1 with each term, until reaching \(n\) in the final term.

Step by step solution

01

Understanding the Binomial Theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} {n\choose k} a^{n-k} b^{k}\), where \({n\choose k}\) represents the number of ways to choose \(k\) items from a set of \(n\) (also called binomial coefficients), \(a^{n-k}\) represents the term involving the variable \(a\) and \(b^{k}\) represents the term involving the variable \(b\).
02

Identifying the pattern of exponents of \(b\)

From the Binomial Theorem, it can be observed that the exponent of \(b\) starts from 0 and increases by 1 in each term till it reaches \(n\) in the final term of the expansion.

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Most popular questions from this chapter

Use the Binomial Theorem to expand and then simplify the result: \(\left(x^{2}+x+1\right)^{3}\) Hint: Write \(x^{2}+x+1\) as \(x^{2}+(x+1)\).

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{n !}{(n-1) !}-\frac{1}{n-1}$$

What is a combination?

Use the formula for \(_{n} C_{r}\) to solve Of the 100 people in the U.S. Senate, 18 serve on the Foreign Relations Committee. How many ways are there to select Senate members for this committee (assuming party affiliation is not a factor in selection)?

What is a permutation?
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