The expansions of \( (a + b)^n \) can be analyzed through the Binomial Theorem, which states:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
Here, \( \binom{n}{k} \) is the binomial coefficient, which counts the number of ways to choose \( k \) elements from a set of \( n \) elements. The key patterns that emerge from these expansions include:
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Coefficients (Binomial Coefficients): The coefficients of each term in the expansion are the binomial coefficients, \( \binom{n}{k} \). These coefficients can be found in Pascal's Triangle, where each number is the sum of the two numbers directly above it. The first few rows are:
- \( n = 0: 1 \)
- \( n = 1: 1 , 1 \)
- \( n = 2: 1 , 2 , 1 \)
- \( n = 3: 1 , 3 , 3 , 1 \)
- \( n = 4: 1 , 4 , 6 , 4 , 1 \)
- and so on.
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Symmetry: The coefficients are symmetric. That is, \( \binom{n}{k} = \binom{n}{n-k} \). This means that the expansion exhibits symmetry around the middle term when \( n \) is even, or around the two middle terms when \( n \) is odd.
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Degree of Terms: Each term in the expansion has a degree that adds up to \( n \). The term \( a^{n-k} b^k \) shows that as \( k \) increases, the power of \( a \) decreases and the power of \( b \) increases, ensuring that the total degree remains \( n \).
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Alternating Signs: If you consider \( (a - b)^n \), the signs of the coefficients alternate. This is due to \( b \) being negative, which means terms with odd \( k \) will have a negative coefficient.
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Sum of Coefficients: The sum of the coefficients of the expansion \( (a + b)^n \) when you set \( a = 1 \) and \( b = 1 \) is \( 2^n \). This implies that the total number of subsets created from \( n \) items (choosing either \( a \) or \( b \) from each item) doubles with each increase in \( n \).
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Recurrence Relation: The binomial coefficients satisfy a recurrence relation given by \( \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} \), which is reflected in Pascal's Triangle.
These patterns are fundamental in combinatorics and have applications in algebra, probability, and various areas of mathematics.
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