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Chapter 7: Problem 405

Find the fifth term of \(\left(2+2 \mathrm{x}^{3}\right)^{17}\).

Short Answer

Expert verified
The fifth term of the expansion of \(\left(2 + 2x^3\right)^{17}\) is \(9,750,240x^{12}\).

Step by step solution

01

Use the binomial theorem to find the fifth term

We will use the binomial theorem for the expansion of \(\left(2 + 2x^3\right)^{17}\): \[(2+2x^3)^{17} = \sum_{k=0}^{17}\binom{17}{k}(2)^{17-k}(2x^3)^k\] We need to find the term with \(k = 4\).
02

Calculate the binomial coefficient

For \(k = 4\), we will calculate the binomial coefficient: \[\binom{17}{4} = \frac{17!}{4!(13)!}\]
03

Calculate factorials

Calculate the factorials: \begin{align*} 17! &= 17 \times 16 \times 15 \times 14 \times 13! \\ 4! &= 4 \times 3 \times 2 \times 1 \\ 13! &= 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \end{align*}
04

Calculate the binomial coefficient

Substitute the factorials in the binomial coefficient formula: \[\binom{17}{4} = \frac{17 \times 16 \times 15 \times 14 \times 13!}{4 \times 3 \times 2 \times 1 \times 13!} = \frac{17 \times 16 \times 15 \times 14}{4 \times 3 \times 2 \times 1} = 1,190\]
05

Find the fifth term

Now, substitute the values of \(k\), \(n\), \(a\), and \(b\) in the binomial theorem formula as well as the binomial coefficient result: \[\binom{17}{4} \cdot 2^{17-4} \cdot (2x^3)^4 = 1190 \cdot 2^{13} \cdot (2x^3)^4\]
06

Simplify the term

Simplify the expression to find the fifth term: \begin{align*} 1190 \cdot 2^{13} \cdot (2x^3)^4 &= 1190 \cdot 8192 \cdot (16x^{12}) \\ &= 9,750,240x^{12} \end{align*} So, the fifth term of the expansion of \(\left(2 + 2x^3\right)^{17}\) is \(9,750,240x^{12}\).

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Most popular questions from this chapter

Find the first five terms of the expansion of \((1+x)^{-2}\).

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