WAEC 2017 · Paper 2 · Q11

(a)
If $f(x) = \displaystyle\int(4x - x^2)\,dx$ and $f(3) = 21$ , find $f(x)$ .
(b)
The second, fourth and eighth terms of an Arithmetic Progression (A.P.) form the first three consecutive terms of a Geometric Progression (G.P.). The sum of the third and fifth terms of the A.P. is 20. Find the: (i) first four terms of the A.P.; (ii) sum of the first ten terms of the A.P.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

The 2nd, 4th and 8th terms of the A.P. are $a + d$ , $a + 3d$ and $a + 7d$ .
They form a G.P., so $(a + 3d)^2 = (a + d)(a + 7d)$ .
Expand: $a^2 + 6ad + 9d^2 = a^2 + 8ad + 7d^2$ , so $2d^2 = 2ad$ and $d(d - a) = 0$ .
An A.P. with $d = 0$ would be constant, so $d = a$ .
The 3rd and 5th terms add to 20: $(a + 2d) + (a + 4d) = 20$ , so $2a + 6d = 20$ .
With $d = a$ : $8a = 20$ , so $a = d = \frac52$ .
The first four terms are $\frac52, 5, \frac{15}{2}, 10$ .

$S_{10} = \dfrac{10}{2}\left[2\left(\frac52\right) + 9\left(\frac52\right)\right]$
$= 5\left(5 + \dfrac{45}{2}\right)$
$= 137.5$ .