Which statement about the simplified binomial expansion of (a - b)^n, where a, b and n are positive integers, is true?
A. All terms of the expansion are positive.
B. All terms in the expansion are negative.
C. The first term in the expansion is positive.
D. The first term in the expansion is negative.
Answer (2)
The binomial expansion of ( a − b ) n is given by ∑ k = 0 n ( k n ) a n − k ( − b ) k .
The sign of each term depends on ( − 1 ) k , which alternates between positive and negative as k increases.
The first term in the expansion (when k = 0 ) is a n , which is positive since a is a positive integer.
Therefore, the first term in the expansion is positive: The first term in the expansion is positive.
Explanation
Understanding the Problem We are given a binomial expansion of the form ( a − b ) n , where a , b , and n are positive integers. We need to determine which statement about the signs of the terms in the simplified binomial expansion is true.
Applying the Binomial Theorem The binomial theorem states that for any positive integer n :
( a − b ) n = k = 0 ∑ n ( k n ) a n − k ( − b ) k Each term in the expansion has the form ( k n ) a n − k ( − b ) k , where ( k n ) is the binomial coefficient, a n − k is a power of a , and ( − b ) k is a power of − b .
Analyzing the Sign of Each Term The sign of each term depends on ( − b ) k = ( − 1 ) k b k . If k is even, then ( − 1 ) k = 1 , so the term is positive. If k is odd, then ( − 1 ) k = − 1 , so the term is negative. Thus, the signs of the terms alternate.
Determining the Sign of the First Term Let's consider the first term in the expansion. This corresponds to k = 0 . The first term is: {n \choose 0} a^{n-0} (-b)^0 = {n \choose 0} a^n (1) = 1 \cdot a^n \\\= a^n Since a is a positive integer, a n is also positive. Therefore, the first term in the expansion is positive.
Conclusion Since the terms alternate in sign, not all terms are positive, and not all terms are negative. The first term is positive, and the terms alternate, so the statement 'The first term in the expansion is positive' is true.
Examples
Binomial expansions are used in probability, statistics, and calculus. For example, they can be used to approximate functions or to model the probability of a certain number of successes in a series of independent trials. Understanding the signs of the terms in a binomial expansion is important in these applications.
The first term in the binomial expansion of ( a − b ) n is positive because it equals a n , where a is a positive integer. Therefore, all terms do not share the same sign, but the first term is indeed positive. The correct answer is C: The first term in the expansion is positive.
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