/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 Find the coefficient of \(w^{198... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 54

Find the coefficient of \(w^{198}\) in the expansion of \((w+3)^{200}\).

Short Answer

Expert verified
The coefficient of \(w^{198}\) in the expansion of \((w+3)^{200}\) is 178200.

Step by step solution

01

Remember the binomial theorem and notation

The binomial theorem states that for any natural number n, and real numbers a and b: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\) Here, we have \(a = w\), \(b = 3\) and \(n = 200\).
02

Identify the term containing \(w^{198}\)

We want to find the term containing \(w^{198}\) in the expansion, i.e., the term where: \(198 = 200 - k\) Solve for k: \(k = 200 - 198\) \(k = 2\)
03

Calculate the binomial coefficient

Now that we have determined that we need to find the coefficient of the term with \(k = 2\), we can compute the binomial coefficient: \(\binom{200}{2}\) = \(\frac{200!}{2!(200-2)!}\) \(\binom{200}{2}\) = \(\frac{200!}{2!(198)!}\)
04

Use the factorial notation to simplify the binomial coefficient

Simplify the expression by calculating the factorials: \(\binom{200}{2}\) = \(\frac{200\times199\times198!}{2\times1(198)!}\) The \(198!\) cancels out, and we are left with: \(\binom{200}{2}\) = \(\frac{200\times199}{2\times1}\)
05

Calculate the binomial coefficient

Now compute the value of the remaining expression: \(\binom{200}{2}\) = \(200\times99\) \(\binom{200}{2}\) = \(19800\)
06

Find the term using the calculated binomial coefficient

We have the binomial coefficient and the exponents of a and b. Substitute them back into the binomial theorem formula to get the term: \(\binom{200}{2}w^{200-2}(3)^2\) Substitute the binomial coefficient: \(19800w^{198}(3)^2\)
07

Finalize the term containing \(w^{198}\)

Simplify and finalize the term containing \(w^{198}\): \(19800w^{198}(9)\) \(178200w^{198}\) The coefficient of \(w^{198}\) in the expansion of \((w+3)^{200}\) is 178200.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate the geometric series. $$ \frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots+\frac{1}{3^{33}} $$

Write the series explicitly and evaluate the sum. $$ \sum_{k=0}^{3} \log \left(k^{2}+2\right) $$

Evaluate \(\lim _{n \rightarrow \infty}\left(1+\frac{3}{n}\right)^{n}\).

In the decimal expansion of \(0.9^{9999}\), how many zeros follow the decimal point before the first nonzero digit?

Use Pascal's triangle to simplify the indicated expression. $$ (3-\sqrt{2})^{6} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.