What is the value of the 5th term of the expansion $(2 x-4)^4$?
Answer (2)
The problem asks for the 5th term of the binomial expansion of ( 2 x − 4 ) 4 .
Recall the binomial theorem: ( a + b ) n = ∑ k = 0 n ( k n ) a n − k b k .
Identify a = 2 x , b = − 4 , n = 4 , and k = 4 for the 5th term.
Calculate the 5th term: ( 4 4 ) ( 2 x ) 4 − 4 ( − 4 ) 4 = 1 ⋅ 1 ⋅ 256 = 256 .
Explanation
Understanding the Problem We are asked to find the 5th term of the binomial expansion of ( 2 x − 4 ) 4 . Let's recall the binomial theorem, which states that for any non-negative integer n and any real numbers a and b :
( a + b ) n = k = 0 ∑ n ( k n ) a n − k b k In our case, we have a = 2 x , b = − 4 , and n = 4 .
Identifying the Correct Term We want to find the 5th term in the expansion. Since the first term corresponds to k = 0 , the second to k = 1 , and so on, the 5th term will correspond to k = 4 .
Applying the Binomial Theorem Now, we can use the binomial theorem to find the 5th term. The term is given by: ( k n ) a n − k b k = ( 4 4 ) ( 2 x ) 4 − 4 ( − 4 ) 4
Simplifying the Expression Let's simplify the expression: ( 4 4 ) ( 2 x ) 4 − 4 ( − 4 ) 4 = ( 4 4 ) ( 2 x ) 0 ( − 4 ) 4 = 1 ⋅ 1 ⋅ ( − 4 ) 4 Since ( 4 4 ) = 1 and ( 2 x ) 0 = 1 , we have: ( − 4 ) 4 = ( − 4 ) × ( − 4 ) × ( − 4 ) × ( − 4 ) = 256
Final Answer Therefore, the 5th term of the expansion ( 2 x − 4 ) 4 is 256.
Examples
Binomial expansions are used in various fields such as probability, statistics, and physics. For example, in probability, they can be used to calculate the probability of a certain number of successes in a series of independent trials. In finance, binomial trees, which are based on binomial expansions, are used to model the price of options.
The 5th term of the expansion ( 2 x − 4 ) 4 is calculated using the binomial theorem, which gives us a value of 256 . This term corresponds to the calculation of ( 4 4 ) ( 2 x ) 0 ( − 4 ) 4 . Therefore, the final answer is 256 .
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