Find the binomial series expansion of
$$
\sqrt{4 - 9x}, \ |x| < \frac{4}{9}
$$
in ascending powers of $x$, up to and including the term in $x^2$ - Edexcel - A-Level Maths Pure - Question 3 - 2018 - Paper 9
Question 3
Find the binomial series expansion of
$$
\sqrt{4 - 9x}, \ |x| < \frac{4}{9}
$$
in ascending powers of $x$, up to and including the term in $x^2$.
Give each coeffici... show full transcript
Worked Solution & Example Answer:Find the binomial series expansion of
$$
\sqrt{4 - 9x}, \ |x| < \frac{4}{9}
$$
in ascending powers of $x$, up to and including the term in $x^2$ - Edexcel - A-Level Maths Pure - Question 3 - 2018 - Paper 9
Step 1
Find the binomial series expansion of $\sqrt{4 - 9x}$
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Answer
To find the binomial series expansion, we start by rewriting the expression:
4−9x=(4(1−49x))1/2=21−49x.
Using the binomial expansion for (1+u)n, where n=21 and u=−49x, we expand:
(1−u)1/2=1−21u+21(21−1)2!u2+…
Substituting our expression gives us:
1−21(−49x)+21(21−1)2(−49x)2
Calculating each term:
First term: 1
Second term: 89x
Third term: −6481⋅2x2=−12881x2
Putting it together:
4−9x≈2(1+89x−12881x2)=2+49x−6481x2.
Thus, the final expansion up to x2 is:
4−9x≈2+49x−6481x2.
Step 2
Use the expansion from part (a), with a suitable value of $x$, to find an approximate value for $\sqrt{310}$
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Answer
To approximate 310, we first note that 310 can be expressed in terms of 4−9x:
Set 4−9x=310,
which gives us:
However, finding a suitable x in the acceptable range ∣x∣<94≈0.444 is essential. We can rewrite 310 as:
