Obtain the binomial expansion of (2-x) (1+1/2x)^8 in ascending powers of x as far as the term in x^3. Use your result to estimate the value of 1.9 times (1.05)^8.
(2-x) (1+1/2 x)^8
= (2-x)[1 + 8(1/2)x)) + 28(1/4)x^2 + 56(1/8)x^3 + ....]
= (2-x)(1 + 4x + 7x^2 + 7x^3 + )
= 2 + 8x + 14x^2 + 14x^3 + ... - x - 4x^2 - 7x^3
= appr 2 + 7x + 10x^2 + 7x^3
so comparing
1.9(1.05)^8 with (2-x)(1 + 1/2 x)^8
x = .1
I get :
2 + .7 + 10(.01) + 7(.001)
= 2 + .7 + .1 + .007
= 2.807
actual answer by calculator is 2.807165343
NOTE:
A rather purely academic exercise.
"Today" we just use a calculator.
Oh, calculating binomial expansions, are we? Let's hop on the clown-car of mathematics!
Using the binomial expansion formula, we can express (2 - x) (1 + (1/2)x)^8 as follows:
1 + (1/2)x - (7/16)x^2 + (7/64)x^3 + ...
Now, to estimate the value of 1.9 times (1.05)^8, we'll substitute x with 0.05 in our binomial expansion:
1 + (1/2)(0.05) - (7/16)(0.05^2) + (7/64)(0.05^3) + ...
Simplifying this lovely mess gives us:
1 + 0.025 - 0.0021875 + 0.000171875 + ...
Adding up the terms up to x^3, we get:
1 + 0.025 - 0.0021875 + 0.000171875 ≈ 1.023984375
So, approximately, 1.9 times (1.05)^8 is around 1.92357153125. Now, don't go clowning around with those decimal places!
To obtain the binomial expansion of (2-x)(1+1/2x)^8, we can use the binomial theorem. The binomial theorem states that for any positive integer n, the expansion of (a+b)^n can be written as:
(a+b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCn * a^0 * b^n
Where nCr represents the binomial coefficient, which can be calculated using the formula nCr = n! / (r! * (n-r)!)
In our case, a = 2-x, and b = (1 + 1/2x). We are interested in expanding up to the term in x^3.
Using the binomial theorem, the expansion up to the term in x^3 would be:
(2-x)(1+1/2x)^8 = (1)(2^8) - (8)(2^7)(x) + (8)(2^6)(x^2) - (8)(2^5)((1/2)^3)(x^3)
Simplifying this expression, we get:
256 - 1024x + 768x^2 - 64x^3
Now, let's use this expression to estimate the value of 1.9 times (1.05)^8.
We can substitute x = 0.05 into the expression and calculate:
1.9 times (1.05)^8 ≈ 1.9 * (256 - 1024(0.05) + 768(0.05)^2 - 64(0.05)^3)
≈ 1.9 * (256 - 1024(0.05) + 768(0.0025) - 64(0.000125))
≈ 1.9 * (256 - 51.2 + 1.92 - 0.00032)
≈ 1.9 * 207.67968
≈ 394.5914
Therefore, the estimate of 1.9 times (1.05)^8 is approximately 394.5914.
To obtain the binomial expansion of (2-x) (1+1/2x)^8, we can use the binomial theorem. The binomial theorem states that for any positive integer n, the expansion of (a+b)^n can be written as the sum of coefficients multiplied by powers of a and b.
In this case, a = 2-x and b = 1/2x. The expansion will have terms up to the power of 8 because we have (1+1/2x)^8. To find the coefficient of each term, we can use the formula for the binomial coefficient:
C(n, k) = n! / (k! * (n-k)!)
Where n is the power of the binomial, and k is the power of a or b in each term.
Let's calculate the expansion up to the term in x^3.
The expansion of (2-x) (1+1/2x)^8, in ascending powers of x, is as follows:
Term 1: C(8, 0) * (2)^8 * (-x)^0 = 256
Term 2: C(8, 1) * (2)^7 * (-x)^1 = -1024x
Term 3: C(8, 2) * (2)^6 * (-x)^2 = 1536x^2
Term 4: C(8, 3) * (2)^5 * (-x)^3 = -1024x^3
To estimate the value of 1.9 times (1.05)^8, we can substitute x = 1.05 into the expansion and evaluate the expression up to the term in x^3:
(2-(1.05)) (1+(1/2 * 1.05))^8
= 0.95 * (1.525)^8
≈ 0.95 * 4.26369
≈ 4.0505095
Therefore, 1.9 times (1.05)^8 is approximately 4.0505095.