Find the designated term of the binomial expansion.
5th term of (4a+2b)^3
This is done without using Pascal's Triangle.
Trick question!
a binomial cubed has only 4 terms.
To find the kth term of a binomial expansion, you can use the formula:
Term(k) = (n choose k) * (a^(n-k)) * (b^k)
In this case, the given expression is (4a + 2b)^3, so n = 3. We want to find the 5th term, so k = 5.
To find (n choose k) without using Pascal's Triangle, you can use the formula:
(n choose k) = n! / (k! * (n-k)!)
Let's plug in the values for n and k:
(n choose k) = 3! / (5! * (3-5)!)
Simplifying further:
(n choose k) = 3! / (5! * (-2!))
Now, calculate the factorial values:
3! = 3 * 2 * 1 = 6
5! = 5 * 4 * 3 * 2 * 1 = 120
-2! is not defined, as the factorial function is only defined for non-negative integers.
Therefore, (n choose k) = 6 / (120 * undefined) = undefined
Since (n choose k) is undefined, the 5th term is also undefined.
In this case, we cannot find the 5th term without using Pascal's Triangle or the factorial function for negative values.