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 (MEI) A Level Maths 1.6 Sequences and Series, covering 1.6.1 Binomial expansion for positive integer n, 1.6.2 Factorial and combinations notation, 1.6.3 Binomial expansion for rational n (A-level only), 1.6.4 Binomial expansion of (a + bx)^n via factoring (A-level only), 1.6.5 Binomial approximating polynomials (A-level only), 1.6.6 What a sequence is; finite and infinite (A-level only), 1.6.7 Generate a sequence by formula or recurrence (A-level only), 1.6.8 A series as a sum of terms (A-level only), 1.6.9 Sigma notation (A-level only), 1.6.10 Increasing, decreasing and periodic sequences (A-level only), 1.6.11 Convergent and divergent sequences (A-level only), 1.6.12 Arithmetic sequences and series (A-level only), 1.6.13 Standard formulae for arithmetic series (A-level only), 1.6.14 Geometric sequences and series (A-level only), 1.6.15 Standard formulae for geometric series (A-level only), 1.6.16 Sum to infinity of a geometric series (A-level only), and 1.6.17 Sequences and series in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.