(a) Write down the first four terms of the binomial expansion of \((2 - \frac{1}{2})^{5}\)

Explanation

(a) From Paschal's triangle, coefficients of the power of 5 are

1, 5, 10, 10, 5, 1

\((2 - \frac{1}{2}x)^{5} = (1)(2^{5}) + 5(2^{4})(-\frac{1}{2}x) + 10(2^{3})(-\frac{1}{2}x)^{2} + 10(2^{2})(-\frac{1}{2}x)^{3} + ...\)

= \(32 - 40x + 20x^{2} - 5x^{3} + ...\)

(b) \((1.99)^{5} = (2 - 0.01)^{5} = (2 - \frac{1}{2}(0.02))^{5}\)

Using the expansion above, \(32 - 40x + 20x^{2} - 5x^{3} + ...\)

\((1.99)^{5} \approxeq 32 - 40(0.02) + 20(0.02)^{2} - 5(0.02)^{3} \)

\(\approxeq 32 - 0.08 + 0.008\)

= \(31.928\)

= \(31.93\) (to 2 decimal place)