Theory question 104
Binomial Theorem(a) Write down the first four terms of the binomial expansion of (2 − ½x)5 in ascending powers of x.
(b) Use your expansion in (a) to find, correct to two decimal places, the value of (1.99)5
Show the model answer
Model answer
(a) 2⁵ + 5(2⁴)(−½x) + 10(2³)(−½x)² + 10(2²)(−½x)³
= 32 − 40x + 20x² − 5x³
(b) 2 − ½x = 1.99 gives x = 0.02
32 − 40(0.02) + 20(0.02)² − 5(0.02)³ = 32 − 0.8 + 0.008 − 0.00004 = 31.20796
(1.99)⁵ ≈ 31.21
Explanation
Use ⁵Cᵣ coefficients with powers of 2 decreasing; then choose x so that 2 − ½x = 1.99.