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Chapter 14: Problem 63

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

Short Answer

Expert verified
In the binomial expansion of \((a+b)^{n}\), the exponents on \(b\) start from 0 and increase by 1 in each subsequent term from left to right, until they equals to \(n\). Simultaneously, the exponents on \(a\) decrease by 1 from \(n\).

Step by step solution

01

Understanding the binomial theorem

The binomial theorem states that: \[(a + b)^n = a^n + {n \choose 1}a^{n-1}b + {n \choose 2}a^{n-2}b^2 + {n \choose 3}a^{n-3}b^3 + ... + b^n\] where \({n \choose k}\) are the binomial coefficients.
02

Writing down the expansions

Let's write down few binomial expansions using the binomial theorem. For \(n=2\), \((a + b)^2 = a^2 + 2ab + b^2\). Here the exponent on \(b\) starts from 0 and increases by 1 until it equals to \(n\). For \(n=3\), \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Here also, the exponent on \(b\) starts from 0 and increases by 1 until it equals to \(n\). This pattern can be generalized to \(n\) terms.
03

Describing the pattern

As we can see from these expansions, the pattern in the exponents of \(b\) in the expansion is as follows: In each term of the expansion, the exponent on \(b\) starts from 0 and increases by 1 when we move from left to right until it equals to \(n\). Also for each \(b^k\) in the term, \(a\) holds the power of \(n-k\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents Pattern
The exponents pattern in the binomial expansion reveals a simple yet elegant symmetry. When expanding equality-oriented expression like equal to equal to equal to equal to equal to equal equal equal equal equal equal equal equal equal to equal to equal to equal equal equal equal equal equal equal to equal to equal inequal inequal

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Most popular questions from this chapter

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 69 and 70 to verify the expansion. $$f_{1}(x)=(x-1)^{3}$$

If \(f(x)=x^{2}+2 x+3,\) find \(f(a+1)\) (Section \(8.1,\) Example 3 )

$$\text { Simplify: } \sqrt[3]{40 x^{4} y^{7}}$$ (Section \(10.3,\) Example 5 )

Will help you prepare for the material covered in the next section. Consider the sequence \(8,3,-2,-7,-12, \ldots .\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the sum of the exponents on the variables in each term.
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