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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q. 33 (page 839)

Find the coefficient of $x^{2}$ in the expansion of ${(2 x + 1)}^{7}$ .

Short Answer

Expert verified

The coefficient of $x^{2}$ is $84 .$

Step by step solution

01

Given information:

Consider the expression ${(2 x + 1)}^{7}$ .

To find the coefficient of $x^{2}$

02

Formulas :

Formula 1: To find a particular term in an expression:

${(x + a)}^{n} = (\begin{matrix} n \\ n - j \end{matrix}) a^{(n - j)} x^{j}$

Formula 2: Binomial Coefficient:

$(\begin{matrix} n \\ j \end{matrix}) = \frac{n!}{j! 脳 (n - j)!}$

03

Step 3:

On comparing ${(2 x + 1)}^{7}$ with ${(x + a)}^{n}$ , we get,

$x = 2 x, a = 1, n = 7$

To find coefficient of $x^{2}$ , $鈬� j = 2$ .

04

Substituting the values in the formula 1:

${(2 x + 1)}^{7} = (\begin{matrix} 7 \\ 7 - 2 \end{matrix}) {(1)}^{(7 - 2)} {(2 x)}^{2}$

05

Simplification:

${(2 x + 1)}^{7} = (\begin{matrix} 7 \\ 5 \end{matrix}) 1^{5} (2^{2} x^{2})$

$= 21 脳 4 x^{2}$

role="math" localid="1647343658235" $[U s e f o r m u l a 2 : (\begin{matrix} 7 \\ 5 \end{matrix}) = \frac{7!}{5! 脳 (7 - 5)!} = \frac{7!}{5! 脳 2!} = \frac{7 脳 6 脳 5!}{5! 脳 (2 脳 1)} = \frac{7 脳 6}{2} = 21]$

$= 84 x^{2}$

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Most popular questions from this chapter

In Problems 17鈥�28, write down the first five terms of each sequence.

$\{t_{n}\} = \{\frac{{(- 1)}^{n}}{(n + 1 (n + 2)}\}$

In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

$a_{1} = 2; a_{n} = 3 + a_{n - 1}$

In Problems 29鈥�36, the given pattern continues. Write down the nth term of a sequence $\{a_{n}\}$ suggested by the pattern.

$\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, . . .$

In Problems 51鈥�66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

$2 + \frac{4}{3} + \frac{8}{9} + . . . .$

In Problems 71-82, find the sum of each sequence.

${鈭� k^{3}}_{k = 4}^{24}$

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