Question

which statement about the simplified binomial expansion of $(a + b^{2})^{n}$, where $n$ is a positive integer, is true?
the exponent of $b$ will always be even.
the exponent of $a$ will always be odd.
the sum of the exponents of $a$ and $b$ will always equal $n$.
the sum of the exponents of $a$ and $b$ will always equal $n - 1$.

Explanation:

Step1: Recall binomial expansion formula

The binomial expansion of $(x+y)^n$ is $\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}$. For $(a + b^2)^n$, substitute $x=a$, $y=b^2$:
$$(a + b^2)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (b^2)^{k} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{2k}$$

Step2: Analyze each option

• Option1: Exponent of $b$ is $2k$, which is even for all integers $k$.
• Option2: Exponent of $a$ is $n-k$. If $n$ is even and $k$ is even, $n-k$ is even, so this is false.
• Option3: Sum of exponents: $(n-k) + 2k = n+k$. Since $k\geq0$, this equals $n$ only when $k=0$, not always. False.
• Option4: Sum is $n+k$, which equals $n-1$ only if $k=-1$, impossible. False.

Answer:

The exponent of $b$ will always be even.