7. (a) Use the binomial expansion, in ascending powers of $x$, to show that
$$
ext{ }
ext{ }
\sqrt{4 - x} = 2 - \frac{1}{4}x + kx^2 + .. - Edexcel - A-Level Maths Pure - Question 8 - 2017 - Paper 2
Question 8
7. (a) Use the binomial expansion, in ascending powers of $x$, to show that
$$
ext{ }
ext{ }
\sqrt{4 - x} = 2 - \frac{1}{4}x + kx^2 + ...
$$
where $k$ is a rationa... show full transcript
Worked Solution & Example Answer:7. (a) Use the binomial expansion, in ascending powers of $x$, to show that
$$
ext{ }
ext{ }
\sqrt{4 - x} = 2 - \frac{1}{4}x + kx^2 + .. - Edexcel - A-Level Maths Pure - Question 8 - 2017 - Paper 2
Step 1
Use the binomial expansion to show that $\sqrt{4 - x} = 2 - \frac{1}{4}x + kx^2 + ...$
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Answer
To begin the binomial expansion of 4−x, we first take out a factor of 4:
4−x=4(1−4x)=21−4x.
Now using the binomial expansion for 1+u where u=−4x, we have:
1−4x=1+21(−4x)+21(21−1)2!1(−4x)2+...
Calculating each term:
The first term: 1
The second term: −81x
The third term: +1281x2
Thus combining these, we have:
4−x=2(1−81x+1281x2+...)=2−41x+kx2+...
From this, we can see that k=641.
Step 2
State, giving a reason, if the expansion is valid for this value of $x$.
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Answer
The expansion is valid for ∣x∣<4, so x=1 can be used. This is because the binomial expansion converges within the radius of convergence, which in this case is defined by the condition that the term ∣4x∣<1.
