WAEC 2022 · Paper 2 · Q9

(a)
Evaluate $\dfrac{7}{2 - \sqrt3} - \dfrac{7}{2 + \sqrt3}$ , leaving the answer in the form $p + q\sqrt n$ , where $p$ , $q$ and $n$ are real numbers.
(b)
The sum of the first and third terms of a Geometric Progression (G.P.) is 20 and the product of the first and fourth terms is 18 times the second term. If the common ratio of the G.P. is positive, find the sum of the first 8 terms.

Worked solution (try it first)

(a)

The common denominator is $(2 - \sqrt3)(2 + \sqrt3) = 4 - 3 = 1$ .
The top is $7(2 + \sqrt3) - 7(2 - \sqrt3) = 14 + 7\sqrt3 - 14 + 7\sqrt3$
$= 14\sqrt3$ .
So the value is $14\sqrt3$ , that is $0 + 14\sqrt3$ .

(b)

Let the first term be $a$ and the common ratio $r$ .
The first and third terms add up to 20: $a + ar^2 = 20$ .
The product of the first and fourth terms is 18 times the second: $a \times ar^3 = 18ar$ .
Divide both sides by $ar$ : $ar^2 = 18$ .
Substitute into the first equation: $a + 18 = 20$ , so $a = 2$ .
Then $2r^2 = 18$ , so $r^2 = 9$ and $r = 3$ (the ratio is positive).
$S_8 = \dfrac{a(r^8 - 1)}{r - 1}$
$= \dfrac{2(6561 - 1)}{2}$
$= 6560$ .

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