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Find the binomial series expansion of √(4−9x), |x| < 49 in ascending powers of x, up to and including the term in x² - Edexcel - A-Level Maths Pure - Question 2 - 2018 - Paper 9
Question 2
Find the binomial series expansion of √(4−9x), |x| < 49 in ascending powers of x, up to and including the term in x². Give each coefficient in its simplest form. (b... show full transcript
Worked Solution & Example Answer:Find the binomial series expansion of √(4−9x), |x| < 49 in ascending powers of x, up to and including the term in x² - Edexcel - A-Level Maths Pure - Question 2 - 2018 - Paper 9
Step 1
Find the binomial series expansion of √(4−9x)
Answer
To find the binomial series expansion for ( \sqrt{4 - 9x} ), we start by rewriting it in a suitable form. We know that:
[ \sqrt{a - b} = a^{1/2}(1 - \frac{b}{a})^{1/2} \text{ where } a = 4 \text{ and } b = 9x ]
Thus, we can express it as:
[ \sqrt{4} (1 - \frac{9x}{4})^{1/2} = 2(1 - \frac{9x}{4})^{1/2} ]
The binomial series expansion for ( (1 + u)^n ) is given by:
[ 1 + nu + \frac{n(n-1)}{2!}u^2 + \frac{n(n-1)(n-2)}{3!}u^3 + \ldots ]
In our case, ( u = -\frac{9x}{4} ) and ( n = \frac{1}{2} ). Therefore, substituting these values:
- First term: ( 1 )
- Second term: ( \frac{1}{2}(-\frac{9x}{4}) = -\frac{9x}{8} )
- Third term: ( \frac{\frac{1}{2}(\frac{1}{2}-1)}{2!}(-\frac{9x}{4})^2 = \frac{1}{2}(-\frac{1}{2})(\frac{81x^2}{16}) \Rightarrow -\frac{81x^2}{64} )
Combining the terms leads to:
[ \sqrt{4 - 9x} = 2 - \frac{9}{8}x - \frac{81}{64}x^2 + O(x^3) ]
Step 2
Use the expansion from part (a), with a suitable value of x, to find an approximate value for √310
Answer
To approximate ( \sqrt{310} ), we can set ( x = 0.1 ) because:
[ 310 = 4 - 9(0.1) \Rightarrow 4 - 0.9 = 3.1 \text{ which is valid as } |0.1| < \frac{4}{9} ]
Using the expansion from part (a) for the selected value:
[ \sqrt{4 - 9(0.1)} \approx 2 - \frac{9}{8}(0.1) - \frac{81}{64}(0.1)^2 ]
Calculating each term:
- ( 2 )
- Second term: ( -\frac{9}{8}(0.1) = -0.1125 )
- Third term: ( -\frac{81}{64}(0.01) = -0.01265625 )
Adding these together:
[ 2 - 0.1125 - 0.01265625 = 1.87484375 ]
Thus, the approximate value for ( \sqrt{310} ) to three decimal places is approximately ( 1.875 ).