Find 7th turm in expension of (x-1/x)*9
AAAaannndd the bot gets it wrong yet again!
The 7th term will be
9C6 x^3(-1/x)^6 = 84/x^3
I apologize for my mistake in the previous response. You are correct that the 7th term in the expansion of (x-1/x)*9 is 84/x^3. Thank you for bringing this to my attention and helping me to improve.
To find the 7th term in the expansion of (x-1/x)*9, we need to expand the expression using binomial expansion.
The general form of the binomial expansion is:
(x + y)^n = C(n, 0) * x^n * y^0 + C(n, 1) * x^(n-1) * y^1 + C(n, 2) * x^(n-2) * y^2 + ... + C(n, r) * x^(n-r) * y^r + ... + C(n, n) * x^0 * y^n
In this case, we can rewrite (x-1/x) as (x^2-1)/x. So the expression becomes:
((x^2-1)/x) * 9
Expanding this expression using binomial expansion, we find:
[(9 * C(2, 0) * x^2 * (-1/x)^0) + (9 * C(2, 1) * x^(2-1) * (-1/x)^1)]
Simplifying further, we get:
[9 * C(2, 0) * x^2 * (-1)^0 + (9 * C(2, 1) * x * (-1))]
Now, let's evaluate the terms:
C(2, 0) = 1
C(2, 1) = 2
So, the expression becomes:
[9 * 1 * x^2 + (9 * 2 * x * (-1))]
Simplifying further, we get:
(9x^2 - 18x)
Now, we need to find the 7th term in this expression. To do that, we use the binomial term formula:
T(r+1) = C(n, r) * x^(n-r) * y^r
Here, r = 7, n = 2, x = 9x^2, and y = -18x.
Using the formula, we can find:
T(8) = C(2, 7) * (9x^2)^(2-7) * (-18x)^7
Simplifying further, we get:
T(8) = 0 (since C(2, 7) = 0)
Therefore, the 7th term in the expansion of (x-1/x)*9 is 0.
The 7th term in the expansion of (x-1/x)*9 can be found using the binomial theorem.
First, we expand (x-1/x) to the 7th power:
(x-1/x)^7 = x^7 - 7x^5 + 21x^3 - 35x + 35/x - 21/x^3 + 7/x^5 - 1/x^7
Next, we multiply this expression by the constant 9:
9(x^7 - 7x^5 + 21x^3 - 35x + 35/x - 21/x^3 + 7/x^5 - 1/x^7)
Expanding this expression and collecting like terms, we find that the 7th term is:
-21x^3
Therefore, the 7th term in the expansion of (x-1/x)*9 is -21x^3.