Find
1.The first 4 terms of the binomial expansion in ascending powers of x of (1+x/4)^8.
2.Use your expansion to estimate (1.025)^8
Hey, I did one, your turn :)
To find the first 4 terms of the binomial expansion in ascending powers of x for (1+x/4)^8, we can use the binomial theorem. The binomial theorem states that (a + b)^n can be expanded as a sum of terms using the binomial coefficients:
(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*b^(n-1) + C(n,n)*a^0*b^n
where C(n,k) is the binomial coefficient, given by C(n,k) = n! / (k!(n-k)!)
Let's apply this to (1+x/4)^8:
First, we need to expand the binomial coefficient C(8,0):
C(8,0) = 8! / (0!(8-0)!)
= 8! / 8!
= 1
So the first term of the expansion is 1*(1)^(8-0)*(x/4)^0 = x^0 = 1.
Next, we expand the binomial coefficient C(8,1):
C(8,1) = 8! / (1!(8-1)!)
= 8! / (1! * 7!)
= 8
So the second term of the expansion is 8*(1)^(8-1)*(x/4)^1 = 8*(x/4) = 2x.
Continuing this process, we can find the third and fourth terms of the expansion:
C(8,2) = 8! / (2!(8-2)!)
= 8! / (2! * 6!)
= 28
So the third term of the expansion is 28*(1)^(8-2)*(x/4)^2 = 28*(x/4)^2 = 7/2 * x^2.
C(8,3) = 8! / (3!(8-3)!)
= 8! / (3! * 5!)
= 56
So the fourth term of the expansion is 56*(1)^(8-3)*(x/4)^3 = 56*(x/4)^3 = 14 * x^3 / 8.
Therefore, the first 4 terms of the binomial expansion in ascending powers of x of (1+x/4)^8 are:
1 + 2x + 7/2 * x^2 + 14 * x^3 / 8.
Now, let's use this expansion to estimate (1.025)^8:
(1.025)^8 ≈ 1 + 2*(1.025) + 7/2 * (1.025)^2 + 14 * (1.025)^3 / 8
Calculating this expression will give us an estimate of (1.025)^8.
To find the first 4 terms of the binomial expansion of (1+x/4)^8, we can use the binomial theorem which states that:
(1+x)^n = C(n,0) x^0 + C(n,1) x^1 + C(n,2) x^2 + C(n,3) x^3 + ...
where C(n,k) represents the binomial coefficient, given by C(n,k) = n! / (k! * (n-k)!)
In our case, we have (1+x/4)^8, so we can substitute n = 8 into our formula.
1. First, let's calculate the binomial coefficients:
C(8,0) = 8! / (0! * (8-0)!) = 1
C(8,1) = 8! / (1! * (8-1)!) = 8
C(8,2) = 8! / (2! * (8-2)!) = 28
C(8,3) = 8! / (3! * (8-3)!) = 56
2. Now, let's write out the first 4 terms of the expansion using the binomial coefficients:
Term 1: C(8,0) (x/4)^0 = 1
Term 2: C(8,1) (x/4)^1 = 8 * (x/4) = 2x
Term 3: C(8,2) (x/4)^2 = 28 * (x/4)^2 = 7x^2/4
Term 4: C(8,3) (x/4)^3 = 56 * (x/4)^3 = 14x^3/16
Therefore, the first 4 terms of the binomial expansion of (1+x/4)^8 are: 1, 2x, 7x^2/4, and 14x^3/16.
To estimate (1.025)^8 using this expansion, we can substitute x = 0.025 into our expansion:
(1.025)^8 ≈ 1 + 2(0.025) + 7(0.025^2)/4 + 14(0.025^3)/16
Calculating the values, we get:
(1.025)^8 ≈ 1 + 0.05 + 0.0034375 + 0.000134765625
(1.025)^8 ≈ 1.0534375
Therefore, (1.025)^8 is approximately equal to 1.0534375.