Find the 5th term of the expression

1) (a-b) ^9

for (a+b)^n

term(r+1) = C(n,r)(a^(n-r) b^r)

so for (a-b)^9
term(5) = C(9,4)(a^5)(-b)^4
= 126a^5 b^4

31

To find the 5th term of the expression (a-b)^9, you can use the binomial theorem.

The binomial theorem states that the expansion of (a + b)^n can be written as the sum of the binomial coefficients multiplied by the corresponding powers of a and b, where n is a non-negative integer.

For our expression (a-b)^9, we can use the binomial theorem with n = 9.

The general formula to find the term of the binomial expansion is given by:
T(k+1) = C(n, k) * a^(n-k) * b^k

Where T(k+1) represents the k+1 th term of the expansion, C(n, k) is the binomial coefficient, a represents the first term (a in this case), b represents the second term (b in this case), n is the exponent (9 in this case), and k is the term number.

Now let's calculate the 5th term:
T(5) = C(9, 5) * a^(9-5) * b^5

The binomial coefficient C(9, 5) can be calculated as C(9, 5) = 9! / (5! * (9-5)!)

Plugging in the values, we get:
T(5) = (9! / (5! * 4!)) * a^4 * b^5

Simplifying further:
T(5) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) * a^4 * b^5

The result is:
T(5) = 126 * a^4 * b^5

Therefore, the 5th term of the expression (a-b)^9 is 126 * a^4 * b^5.