Find the term independent of x in the expansion of
100
4
3
2
x
x
The given expression is:
(1 - x/4)^100
Using the binomial theorem, we can expand this as:
∑(100 choose k)*(1)^(100-k)*(-x/4)^k
= ∑(100 choose k)*(-1)^k*(x/4)^k
The term independent of x corresponds to k = 0, since any power of x greater than 0 will have a factor of x. Therefore, the term independent of x is:
(100 choose 0)*(-1)^0*(1/4)^0 = 1.
So the answer is 1.
To find the term independent of x in the expansion of (1 - x)^100, we can use the binomial theorem. The binomial theorem states that for any positive integer n and any real numbers a and b,
(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n,
where C(n, k) represents the binomial coefficient.
In this case, a = 1 and b = -x. The expansion we are interested in is (1 - x)^100.
Using the binomial theorem, we can see that the powers of x decrease from a^100 * b^0 (in the first term) to a^0 * b^100 (in the last term). The term independent of x will occur when the exponent of x is 0, which means we need to find the term with k = 100.
Using the binomial coefficient C(n, k) = n! / (k!(n-k)!), we can calculate the coefficient of the term with k = 100:
C(100, 100) = 100! / (100!(100-100)!) = 100! / (100! * 0!) = 1.
Therefore, the term independent of x in the expansion of (1 - x)^100 is 1 * 1^0 * (-x)^100, which simplifies to -x^100.