Math Problem Statement
Given that in the expansion of
(4x2−3x)n
, the
5th
term, in descending powers of
x
, is a constant, find the value of
n
.
Solution
To find the value of in the expansion of such that the 5th term is a constant, we need to determine the term which corresponds to the 5th position in the binomial expansion.
The general term in the binomial expansion of is given by: where and .
To find the 5th term (which is actually in binomial terms, because indexing starts from ), we set up the term:
Simplifying :
= \binom{n}{4} (4^{n-4} x^{2(n-4)}) ((-3)^4 x^4) = \binom{n}{4} 4^{n-4} (-3)^4 x^{2(n-4) + 4} = \binom{n}{4} 4^{n-4} 81 x^{2n-8 + 4} = \binom{n}{4} 4^{n-4} 81 x^{2n - 4}$$ We want this term to be a constant. For it to be a constant, the power of $$x$$ must be zero: $$2n - 4 = 0$$ Solving for $$n$$: $$2n - 4 = 0 \\ 2n = 4 \\ n = 2$$ Therefore, the value of $$n$$ is: $$\boxed{2}$$ Would you like more details or have any questions about this solution? Here are 5 related questions you might want to ask: 1. How do you determine the general term in a binomial expansion? 2. What are the steps to find a specific term in the binomial expansion? 3. How do you handle binomial expansions with negative exponents? 4. What is the binomial theorem and how is it applied? 5. Can you show an example of finding a specific term in a different binomial expansion? **Tip:** When dealing with binomial expansions, always remember to consider the indexing properly and simplify each term step-by-step to avoid mistakes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Binomial theorem
Binomial expansion
Exponent rules
Formulas
Binomial theorem formula
General term formula in binomial expansion
Theorems
Binomial theorem
Suitable Grade Level
High School