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Chapter 14: Problem 65

How do you determine how many terms there are in a binomial expansion?

Short Answer

Expert verified
For a binomial expansion of \((a + b)^n\), the number of terms in the expanded form is always \(n + 1\). So for \((a + b)^7\), there would be 8 terms.

Step by step solution

01

Understanding Binomial Expansion

A binomial is an algebraic expression which consists of two terms, like \(a + b\). The expansion of a binomial expression raised to a certain power, such as \((a + b)^n\), is known as binomial expansion. The structure of a binomial expansion comes from Binomial Theorem, which states that \((a + b)^n = a^n + {n \choose 1}a^{n-1}b + {n \choose 2}a^{n-2}b^2 + {n \choose 3}a^{n-3}b^3 + ...\)
02

Deducing the number of terms

In Binomial Theorem, the exponent 'n' indicates the number of terms in the expanded form when 'n' is a nonnegative integer. Regardless of what the terms \(a\) and \(b\) are, the number of terms in a binomial expansion of the format \((a + b)^n\) is always \(n + 1\). This comes from the fact that in a binomial expansion, terms range from \(0\) to \(n\), which in total gives \(n + 1\) terms.
03

Applying this rule

Let's say we're asked to determine the number of terms in the binomial expansion of \((a + b)^7\), using the rule established earlier, there will be \(7 + 1 = 8\) terms in the expansion.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used a formula to find the sum of the infinite geometric series \(3+1+\frac{1}{3}+\frac{1}{9}+\cdots\) and then checked my answer by actually adding all the terms.

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$1,4,9,16, \dots$$

Will help you prepare for the material covered in the next section. Use the formula \(a_{n}=4+(n-1)(-7)\) to find the eighth term of the sequence \(4,-3,-10, \ldots\)

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