Determine the binomial expansion of $\left(1+\frac{1}{4} x\right)^{10}$ in ascending powers of x up to and including term $x^3$; hence use the expansion to determine the value of $\left(\frac{41}{40}\right)^{10}$
Answer (2)
Find the binomial expansion of ( 1 + 4 1 x ) 10 up to the x 3 term: 1 + 2.5 x + 2.8125 x 2 + 1.875 x 3 .
Determine the value of x such that 1 + 4 1 x = 40 41 , which gives x = 0.1 .
Substitute x = 0.1 into the expansion: 1 + 2.5 ( 0.1 ) + 2.8125 ( 0.1 ) 2 + 1.875 ( 0.1 ) 3 .
Calculate the approximate value: 1.2800 .
Explanation
Understanding the Problem We are asked to find the binomial expansion of ( 1 + 4 1 x ) 10 up to the term x 3 and then use this expansion to approximate the value of ( 40 41 ) 10 .
Applying the 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 = ( 0 n ) a n b 0 + ( 1 n ) a n − 1 b 1 + ( 2 n ) a n − 2 b 2 + ⋯ + ( n n ) a 0 b n In our case, a = 1 , b = 4 1 x , and n = 10 . We want to find the terms up to x 3 .
Calculating Binomial Coefficients The binomial coefficients are given by ( k n ) = k ! ( n − k )! n ! . Let's calculate the first few coefficients: ( 0 10 ) = 1 ( 1 10 ) = 1 ! 9 ! 10 ! = 10 ( 2 10 ) = 2 ! 8 ! 10 ! = 2 10 × 9 = 45 ( 3 10 ) = 3 ! 7 ! 10 ! = 3 × 2 × 1 10 × 9 × 8 = 120
Writing the Binomial Expansion Now, let's write down the binomial expansion up to the x 3 term: ( 1 + 4 1 x ) 10 = ( 0 10 ) ( 1 ) 10 ( 4 1 x ) 0 + ( 1 10 ) ( 1 ) 9 ( 4 1 x ) 1 + ( 2 10 ) ( 1 ) 8 ( 4 1 x ) 2 + ( 3 10 ) ( 1 ) 7 ( 4 1 x ) 3 + ⋯ = 1 + 10 ( 4 1 x ) + 45 ( 16 1 x 2 ) + 120 ( 64 1 x 3 ) + ⋯ = 1 + 4 10 x + 16 45 x 2 + 64 120 x 3 + ⋯ = 1 + 2.5 x + 2.8125 x 2 + 1.875 x 3 + ⋯
Finding the Value of x We want to approximate ( 40 41 ) 10 using this expansion. We need to find the value of x such that 1 + 4 1 x = 40 41 .
4 1 x = 40 41 − 1 = 40 1 x = 40 4 = 10 1 = 0.1
Approximating the Value Now, substitute x = 0.1 into the binomial expansion: ( 40 41 ) 10 ≈ 1 + 2.5 ( 0.1 ) + 2.8125 ( 0.1 ) 2 + 1.875 ( 0.1 ) 3 = 1 + 0.25 + 2.8125 ( 0.01 ) + 1.875 ( 0.001 ) = 1 + 0.25 + 0.028125 + 0.001875 = 1.2800
Final Answer Therefore, the approximate value of ( 40 41 ) 10 using the binomial expansion is 1.2800.
Examples
Binomial expansion is a powerful tool used in various fields such as physics, engineering, and finance. For instance, in finance, it can be used to approximate the value of compound interest over a period. In physics, it can be used to simplify complex equations involving small changes in variables. Understanding binomial expansion allows for quick and accurate approximations in these real-world scenarios, saving time and resources.
The binomial expansion of ( 1 + 4 1 x ) 10 up to the term with x 3 is 1 + 2.5 x + 2.8125 x 2 + 1.875 x 3 . By setting 1 + 4 1 x = 40 41 and finding x = 0.1 , we approximate ( 40 41 ) 10 as approximately 1.2800 .
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