WAEC 2009 · Paper 2 · Q4

Sequences, series & binomial expansion

(a)
Write down the first three terms of the binomial expansion of $(1 + 3x)^{\frac13}$ in ascending powers of $x$ .
(b)
Use the expansion in (a) to find, correct to three decimal places, the value of $28^{\frac13}$ .

Worked solution (try it first)

(a)

$(1 + a)^n = 1 + na + \dfrac{n(n - 1)}{2!}a^2 + \ldots$ with $n = \frac13$ and $a = 3x$ .
Second term: $\frac13(3x) = x$ .
Third term: $\dfrac{\frac13\left(-\frac23\right)}{2}(3x)^2 = -\frac19 \times 9x^2$
$= -x^2$ .
So $(1 + 3x)^{\frac13} = 1 + x - x^2 + \ldots$

(b)

Write $28 = 27\left(1 + \frac{1}{27}\right)$ , so $28^{\frac13} = 3\left(1 + \frac{1}{27}\right)^{\frac13}$ .
Match $3x = \frac{1}{27}$ : $x = \frac{1}{81}$ .
$28^{\frac13} \approx 3\left(1 + \frac{1}{81} - \frac{1}{6561}\right)$
$= 3(1 + 0.012346 - 0.000152)$
$= 3(1.012193)$ .
$= 3.03658$ , which is $3.037$ to 3 decimal places.

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