Using Pascal’s Triangle, what is the third term in the expansion of (a+5)5
when the expanded polynomial is written in standard form?(1 point)
Responses
10a3
10 A cubed
250a3
250 A cubed
250a2
250 A squared
25a3
]
250a3
To find the third term in the expansion of (a + 5)^5, we can use Pascal's Triangle.
First, let's write out the expansion using the binomial theorem:
(a + 5)^5 = 1*(a^5)*(5^0) + 5*(a^4)*(5^1) + 10*(a^3)*(5^2) + 10*(a^2)*(5^3) + 5*(a^1)*(5^4) + 1*(a^0)*(5^5)
Now, let's simplify the expression:
1*(a^5)*(5^0) = a^5
5*(a^4)*(5^1) = 5a^4 * 5 = 25a^4
10*(a^3)*(5^2) = 10a^3 * 25 = 250a^3
10*(a^2)*(5^3) = 10a^2 * 125 = 1250a^2
5*(a^1)*(5^4) = 5a * 625 = 3125a
1*(a^0)*(5^5) = 625
So, the third term in the expansion is 250a^3.
Therefore, the correct answer is option 2: 250a^3.
To find the third term in the expansion of (a+5)^5, we can use Pascal's Triangle and the binomial theorem. The general formula for the nth term in the expansion of (a+b)^n is given by:
C(n-1, r-1) * a^(n-r) * b^r
Where C(n-1, r-1) is the binomial coefficient, also known as the "choose" function, which represents the number of ways to choose r objects from a set of n objects.
In this case, we have (a+5)^5, so our values are:
n = 5 (the exponent)
a = a (the base)
b = 5 (the constant term)
To find the third term, we need to determine the value of r. In Pascal's Triangle, each term represents a row, and each number in a row represents a specific term in the expansion. The triangle starts with the number 1, and each subsequent number is the sum of the two numbers above it.
The third term corresponds to the third number in Pascal's Triangle for row 5. The first number in row 5 is 1, the second number is 5, and the third number is 10.
So, the binomial coefficient for our third term is C(5-1, 3-1) = C(4, 2) = 6.
Now, we can substitute the values into the formula:
6 * a^(5-3) * 5^3 = 6 * a^2 * 125 = 750a^2
Therefore, the third term in the expansion of (a+5)^5, when written in standard form, is 750a^2.