Use binomial expansions to show that
1+5x1−x≈1+3x−32x2\displaystyle \sqrt{\frac{1 + 5x}{1 - x}} \approx 1 + 3x - \frac{3}{2}x^21−x1+5x≈1+3x−23x2
for small values of xxx.
A student substitutes x=12\displaystyle x = \frac{1}{2}x=21 into the expansion to find an estimate for 7\sqrt{7}7. Give a reason why the student should not use x=12\displaystyle x = \frac{1}{2}x=21.
Substitute x=19\displaystyle x = \frac{1}{9}x=91 into 1+5x1−x≈1+3x−32x2\displaystyle \sqrt{\dfrac{1 + 5x}{1 - x}} \approx 1 + 3x - \frac{3}{2}x^21−x1+5x≈1+3x−23x2 to obtain an approximation for 7\sqrt{7}7. Give your answer as a fraction in its simplest form.
308 exam-style questions on OCR A Level Maths 1.4 Sequences and Series, covering 1.4.1 Binomial expansion for positive integer n, 1.4.2 Link to binomial probabilities, 1.4.3 Binomial expansion for rational n (A-level only), 1.4.4 Validity of the expansion (A-level only), 1.4.5 Sequences (A-level only), 1.4.6 Increasing, decreasing and periodic sequences (A-level only), 1.4.7 Sigma notation (A-level only), 1.4.8 Arithmetic sequences and series (A-level only), 1.4.9 Geometric sequences and series (A-level only), 1.4.10 Sum to infinity of a geometric series (A-level only), and 1.4.11 Modelling with sequences and series (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.