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Chapter 14: Problem 45

Find the indicated term in the binomial series. $$(p-j)^{15}, j^{11} \text { -term }$$

Short Answer

Expert verified
-1365p^4j^{11}

Step by step solution

01

Identify the General Term of Binomial Expansion

The general term in the expansion of \( (a+b)^n \) is given by the formula \[ T_{k+1} = {n\choose k} a^{n-k} b^k \] where \( T_{k+1} \) is the \( k+1 \)th term, \( {n\choose k} \) is the binomial coefficient, \( a \) and \( b \) are the terms in the binomial, \( n \) is the exponent, and \( k \) is the specific term number minus one.
02

Substitute the Given Values into the General Formula

For the binomial \( (p-j)^{15} \) and the term \( j^{11} \) we want to find, substitute \( a=p \) and \( b=-j \), \( n=15 \) and \( k=11 \) into the formula from Step 1 to find the coefficient of \( j^{11} \) term.
03

Calculate the Binomial Coefficient

Compute the binomial coefficient \( {15\choose 11} \) which is \( \frac{15!}{11!(15-11)!} = \frac{15!}{11!4!} \) to get the coefficient for the \( j^{11} \) term.
04

Find the Exponent of the First Term

Since the exponent of \( j \) is 11 and the total exponent is 15, the exponent of \( p \) would be \( 15-11=4 \) for the \( j^{11} \) term.
05

Write Out the Term with Calculated Coefficient and Exponents

The coefficient obtained in Step 3, along with the exponents from Step 4, gives us the \( j^{11} \) term of the binomial expansion: \[ {15\choose 11}p^4(-j)^{11} = \frac{15!}{11!4!}p^4(-1)^{11}j^{11} \]
06

Simplify the Term

Simplify the expression to find the coefficient: \[ {15\choose 11}p^4(-j)^{11} = 1365p^4(-1)^{11}j^{11} = -1365p^4j^{11} \] since \( {15\choose 11} = 1365 \) and \( (-1)^{11} = -1 \)

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Key Concepts

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

Binomial Expansion
The binomial expansion describes the algebraic expansion of powers of a binomial. The binomial theorem states that a binomial raised to a positive integer, say \( a + b \)^n, can be expanded into a sum of terms. Each term is a product of a binomial coefficient and the two variables \( a \) and \( b \) raised to varying powers. For example:\
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Most popular questions from this chapter

The infinite set of numbers \(5,7,9,11, \ldots\) is an arithmetic sequence. It progresses by adding 2 to one term to get the next term. What does the tenth term equal? How many 2 s would you have to add to the first term, \(5,\) to get the tenth term? How could you get the tenth term quickly? Find the 100 th term quickly.

Write out the terms of the partial sum and add them. $$S_{6}=\sum_{n=1}^{6} 3^{n}$$

Write out the terms of the partial sum and add them. $$S_{7}=\sum_{n=1}^{7} n^{2}$$

A series has a partial sum $$ S_{6}=\sum_{n=1}^{6} 5 \cdot 3^{n-1} $$ a. Write out the terms of the partial sum How you can tell that the series is geometric? b. Evaluate \(S_{6}\) three ways: numerically, by adding the six terms; algebraically, by using the pattern (first term)(fraction involving \(r\) ) and numerically, by storing the formula for \(t_{n}\) in the \(y=\) menu and using your grapher program. Are the answers the same? c. Evaluate \(S_{20}\) for this series. Which method did you use?

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