How do I find the fourth term in the expansion of (a +b)^5?

Question

How do I find the fourth term in the expansion of (a +b)^5?

Answer 1

Your old friend pascal can help you! He has a useful triangle that helps you evaluate the terms in expansions. you start with 1, then 11, then you add 1 to the ends and add the two previous terms above it together to get the middle terms.

1

1 1 (a + b)

1 2 1 (a + b)^2

1 3 3 1 (a + b)^3

1 4 6 4 1 (a + b)^4

1 5 10 10 5 1 (a + b)^5

I only did it for what we need. With this, it says the 4th term should be 10, or 10 a^2b^3. To find what 'a' and 'b' are, you just could how many terms are to the right for 'a' and to the left for 'b'.

I hope this helps!

Answer 2

Well since the exponent is 5, there is going to be 6 terms. The easy way to find the coefficient to each term is the first coefficient is 5C0, then 5C1 all the way to 5C5. Therefore, the fourth term's coefficient is 5C3, or 10.

Now take the first part of the binomial, in this case, a. In the first term it's a^5, then a^4, all the way down to a^0, making the fourth term a^2. For the second part of the binomial, b, it is just the opposite, making the fourth term b^3. To check, the exponents for the first and second parts of the binomial should always add up to the exponent over the entire binomial.

Now all you have to do is multiply all your answers together, leaving you with 10a^2b^3