Find the first 4 terms of the binomial expansion, in ascending powers of $x$, of $\left(1+\frac{x}{4}\right)^8$ giving each term in its simplest form.
Use your expansion to estimate the value of $(1.025)^8$ giving your answer to 4 decimal places.
Answer (2)
Find the first four terms of the binomial expansion of ( 1 + 4 x ) 8 : 1 + 2 x + 4 7 x 2 + 8 7 x 3 .
Set 1 + 4 x = 1.025 and solve for x , which gives x = 0.1 .
Substitute x = 0.1 into the expansion: 1 + 2 ( 0.1 ) + 4 7 ( 0.1 ) 2 + 8 7 ( 0.1 ) 3 = 1.218375 .
Round the result to 4 decimal places: 1.2184 .
Explanation
Problem Analysis We are asked to find the first four terms of the binomial expansion of ( 1 + 4 x ) 8 and then use this expansion to estimate the value of ( 1.025 ) 8 to four decimal places.
Binomial Theorem The binomial theorem states that for any positive integer n and any real numbers a and b :
( a + b ) n = k = 0 ∑ n ( k n ) a n − k b k In our case, a = 1 , b = 4 x , and n = 8 . We need to find the terms for k = 0 , 1 , 2 , 3 .
Calculating the Terms Let's calculate the first four terms:
Term 1 ( k = 0 ): ( 0 8 ) ( 1 ) 8 ( 4 x ) 0 = 1 × 1 × 1 = 1
Term 2 ( k = 1 ): ( 1 8 ) ( 1 ) 7 ( 4 x ) 1 = 8 × 1 × 4 x = 2 x
Term 3 ( k = 2 ): ( 2 8 ) ( 1 ) 6 ( 4 x ) 2 = 28 × 1 × 16 x 2 = 16 28 x 2 = 4 7 x 2
Term 4 ( k = 3 ): ( 3 8 ) ( 1 ) 5 ( 4 x ) 3 = 56 × 1 × 64 x 3 = 64 56 x 3 = 8 7 x 3
So, the first four terms of the binomial expansion are 1 + 2 x + 4 7 x 2 + 8 7 x 3 .
Estimating (1.025)^8 Now, we need to estimate ( 1.025 ) 8 using our expansion. We set 1 + 4 x = 1.025 . Solving for x :
4 x = 0.025
x = 4 × 0.025 = 0.1 Substitute x = 0.1 into the first four terms of the binomial expansion: 1 + 2 ( 0.1 ) + 4 7 ( 0.1 ) 2 + 8 7 ( 0.1 ) 3 = 1 + 0.2 + 4 7 ( 0.01 ) + 8 7 ( 0.001 ) = 1 + 0.2 + 0.0175 + 0.000875 = 1.218375 Rounding to 4 decimal places, we get 1.2184 .
Final Answer Therefore, the first four terms of the binomial expansion of ( 1 + 4 x ) 8 are 1 + 2 x + 4 7 x 2 + 8 7 x 3 , and the estimated value of ( 1.025 ) 8 to 4 decimal places is 1.2184 .
Examples
Binomial expansion is used in various fields such as physics, engineering, and finance. For example, in finance, it can be used to approximate the value of an investment over time, considering small rates of return. Imagine you invest $1000 with an annual interest rate of 2.5%. Using the binomial expansion, you can estimate the value of your investment after 8 years, assuming the interest is compounded annually. This provides a quick and reasonably accurate estimate without needing to calculate each year's compounding individually.
The first four terms of the binomial expansion of ( 1 + 4 x ) 8 are 1 + 2 x + 4 7 x 2 + 8 7 x 3 . Using this to estimate ( 1.025 ) 8 results in approximately 1.2184 .
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