Find the indicated terms in the expansion of the given binomial.
The first three terms in the expansion of
(x+1/x)^40
first term
second term
third term
x^40 + 40 x^38 + 780 x^36
https://socratic.org/questions/how-do-you-find-the-fourth-term-of-a-b-8
as an example
an expansion of ... (a + b)^k
for the nth term
... (k)C(n-1) a^(k-n+1) b^(n-1)
Thanks
Scott How do you get 780x^36 for the 3rd term? Thx
To find the indicated terms in the expansion of the given binomial (x+1/x)^40, we can use the binomial theorem. The binomial theorem states that for any real number x and positive integer n, the expansion of (x+y)^n can be written as the sum of the terms in the form C(n,k) * x^(n-k) * y^k, where C(n,k) is the binomial coefficient.
In this case, we have (x+1/x)^40. By applying the binomial theorem, the first three terms can be calculated as follows:
First term: To find the first term, we use k = 0 in the formula. Therefore, the first term is C(40, 0) * x^(40-0) * (1/x)^0, which simplifies to 1 * x^40 = x^40.
Second term: To find the second term, we use k = 1 in the formula. Therefore, the second term is C(40, 1) * x^(40-1) * (1/x)^1, which simplifies to 40 * x^39 * (1/x)^1 = 40x^39.
Third term: To find the third term, we use k = 2 in the formula. Therefore, the third term is C(40, 2) * x^(40-2) * (1/x)^2, which simplifies to (40 * 39 / (2 * 1)) * x^38 * (1/x)^2 = 780x^38.
Therefore, the first three terms in the expansion of (x+1/x)^40 are:
First term: x^40
Second term: 40x^39
Third term: 780x^38.