In the expansion of (a - b)^n, how does the sign of the terms alternate?

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Question: In the expansion of (a - b)^n, how does the sign of the terms alternate?

Options:

Correct Answer: The signs alternate starting with negative.

Solution:

In the expansion of (a - b)^n, the signs alternate starting with negative due to the negative sign in front of b.

In the expansion of (a - b)^n, how does the sign of the terms alternate?

In the expansion of (a - b)^n, how does the sign of the terms alternate?

Step 1: Understand the expression (a - b)^n. This means we are raising the expression (a - b) to the power of n.
Step 2: Recognize that when we expand (a - b)^n using the Binomial Theorem, we will get multiple terms.
Step 3: The Binomial Theorem states that (a - b)^n = Σ (n choose k) * a^(n-k) * (-b)^k, where k goes from 0 to n.
Step 4: Notice that (-b)^k means that the sign of b will change depending on the value of k.
Step 5: When k is even, (-b)^k is positive, and when k is odd, (-b)^k is negative.
Step 6: Therefore, the first term (when k=0) is positive, the second term (when k=1) is negative, the third term (when k=2) is positive, and so on.
Step 7: This creates an alternating pattern of signs in the expansion, starting with a positive term when k=0.

Binomial Expansion – The expansion of (a - b)^n involves using the binomial theorem, which states that (x + y)^n can be expanded into a series of terms involving combinations of x and y raised to various powers.
Sign Alternation – In the expansion of (a - b)^n, the sign of each term alternates due to the negative sign associated with b, leading to a pattern of positive and negative terms.