QuestionFind the 2 nd term in the expansion of $(x+5 y)^{6}$ in simplest form.

Studdy Solution

STEP 1

What is this asking? We need to find the second term in the expansion of a binomial expression. Watch out! Remember that the second term corresponds to $n=1$ in the binomial theorem, not $n=2$ !

STEP 2

1. Binomial Theorem Setup
2. Calculate the Binomial Coefficient
3. Calculate the Term

STEP 3

Alright, let's start by remembering the amazing binomial theorem!
It says that for any positive integer $n$ , we have $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k.$ This might look intimidating, but it's just a fancy way of writing out the expansion of $(a+b)^n$ .

STEP 4

In our case, we have $a = x$ , $b = 5y$ , and $n = 6$ .
We want the second term, which means we need the term with $k = 1$ (remember, we start counting from $k=0$ !).

STEP 5

Let's calculate the binomial coefficient $\binom{n}{k}$ , which in our case is $\binom{6}{1}$ .
Remember, this is calculated as: $\binom{n}{k} = \frac{n!}{k!(n-k)!}.$ So, for us: $\binom{6}{1} = \frac{6!}{1!(6-1)!} = \frac{6!}{1!5!} = \frac{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(1)(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)} = \frac{6}{1} = 6.$ Our binomial coefficient is 6!

STEP 6

Now, let's plug everything into the binomial theorem formula for the $k=1$ term: $\binom{n}{k} a^{n-k} b^k = \binom{6}{1} x^{6-1} (5y)^1.$ We already know $\binom{6}{1} = 6$ , so we have: $6 \cdot x^5 \cdot (5y).$

STEP 7

Let's simplify: $6 \cdot x^5 \cdot 5y = 6 \cdot 5 \cdot x^5 \cdot y = 30x^5y.$

STEP 8

The second term in the expansion of $(x+5y)^6$ is $30x^5y$ .

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