Use binomial expansion to show that 1+2x1−x≈1+32x+38x2\displaystyle \sqrt{\frac{1+2x}{1-x}} \approx 1 + \frac{3}{2}x + \frac{3}{8}x^21−x1+2x≈1+23x+83x2
A student substitutes x=23\displaystyle x = \frac{2}{3}x=32 into the expansion to find an estimate for 7\sqrt{7}7. Give a reason why the student should not use x=23\displaystyle x = \frac{2}{3}x=32
Substitute x=15\displaystyle x = \frac{1}{5}x=51 into 1+2x1−x=1+32x+38x2\displaystyle \sqrt{\frac{1+2x}{1-x}} = 1 + \frac{3}{2}x + \frac{3}{8}x^21−x1+2x=1+23x+83x2 to obtain an approximation for 7\sqrt{7}7. Give your answer as a fraction in its simplest form.
408 exam-style questions on Edexcel A Level Maths Binomial Expansion, covering 4.1 Expanding (1 + x)^n, 4.2 Expanding (a + bx)^n, and 4.3 Using Partial Fractions. Each one has a worked solution and a mark scheme showing where the marks go.