Which statement about the simplified binomial expansion of $\left(a+b^2\right)^n$, where $n$ is a positive integer, is true?
A. The exponent of $b$ will always be even.
B. The exponent of a will always be odd.
C. The sum of the exponents of $a$ and $b$ will always equal $n$.
D. The sum of the exponents of $a$ and $b$ will always equal $n-1$.
Answer (1)
The binomial theorem expands ( a + b 2 ) n into ∑ k = 0 n ( k n ) a n − k b 2 k .
The exponent of b is 2 k , which is always even.
The exponent of a is n − k , which can be even or odd depending on n and k .
The sum of the exponents of a and b is n + k , which is not always equal to n or n − 1 .
Therefore, the statement 'The exponent of b will always be even' is true. \boxed{\text{The exponent of b will always be even.}}
Explanation
Understanding the Problem We are given the binomial expansion of ( a + b 2 ) n , where n is a positive integer. We need to determine which of the given statements about the simplified binomial expansion is true.
Applying the Binomial Theorem The binomial theorem states that ( x + y ) n = ∑ k = 0 n ( k n ) x n − k y k . Applying this to our expression, we have ( a + b 2 ) n = ∑ k = 0 n ( k n ) a n − k ( b 2 ) k = ∑ k = 0 n ( k n ) a n − k b 2 k .
Analyzing the Statements Let's analyze each statement:
The exponent of b will always be even. The general term in the expansion is ( k n ) a n − k b 2 k . The exponent of b is 2 k . Since k is an integer, 2 k is always an even number. Thus, the exponent of b will always be even. This statement is true.
The exponent of a will always be odd. The exponent of a is n − k . Since n can be either even or odd, and k ranges from 0 to n , n − k can be either even or odd. For example, if n = 2 and k = 0 , then n − k = 2 , which is even. If n = 2 and k = 1 , then n − k = 1 , which is odd. Thus, the exponent of a is not always odd. This statement is false.
The sum of the exponents of a and b will always equal n . The sum of the exponents is ( n − k ) + 2 k = n + k . Since k varies from 0 to n , the sum n + k is not always equal to n . For example, if n = 2 and k = 1 , the sum is 2 + 1 = 3 , which is not equal to n = 2 . This statement is false.
The sum of the exponents of a and b will always equal n − 1 . As we found in statement 3, the sum of the exponents is n + k . This is not always equal to n − 1 . For example, if n = 2 and k = 0 , the sum is 2 + 0 = 2 , which is not equal to n − 1 = 1 . This statement is false.
Conclusion Therefore, only the first statement is true.
Examples
Binomial expansions are used in probability calculations, such as determining the likelihood of different outcomes in a series of independent trials. For instance, if you're analyzing the probability of getting a certain number of heads when flipping a biased coin multiple times, the binomial theorem helps expand and calculate those probabilities accurately. Understanding the properties of binomial expansions, like the even exponent of b in ( a + b 2 ) n , simplifies these calculations and provides insights into the distribution of probabilities.
