(a) Write down the binomial expansion of ((2 – x)^{5}) in ascending powers of x.
(b) Use your expansion in (a) to evaluate ((1.98)^{5}) correct to four decimal places.
Explanation
(a) ((2 – x)^{5} = 2^{5} + 5[2^{4}(-x)^{1}] + 10[2^{3}(-x)^{2}] + 10[2^{2}(-x)^{3}] + 5[2^{1}(-x)^{4}] + (-x)^{5})
= (32 – 80x + 80x^{2} – 40x^{3} + 10x^{4} – x^{5})
(b) ((1.98)^{5} = (2 – 0.02)^{5})
= (32 – 80(0.02) + 80(0.02)^{2} – 40(0.02)^{3} + 10(0.02)^{4} – (0.02)^{5})
= (32 – 1.6 + 0.032 – 0.00032 + …)
(approxeq 30.4317) (4 d.p)