Question

The sums of the first n, 2n and 3n terms of an AP are respectively denoted by x, y and z. Find the value of $\frac{z}{y-x}$ [4 MARKS]

Answer 1

Formula: 1 Mark
Application: 2 Marks
Answer: 1 Mark

Let a and d be the first term and the common difference of the AP. We have:

$x=S_n=\frac{n}{2}(2a+(n-1\left)d\right)$

$y=S_{2n}=\frac{2n}{2}(2a+(2n-1\left)d\right)$

$z=S_{3n}=\frac{3n}{2}(2a+(3n-1\left)d\right)$

$y-x=n(2a+(2n-1)d-\frac{n}{2}(2a+(n-1)d\left)\right)$

$=n(a+(\frac{3n}{2}-\frac{1}{2})d)$

$=\frac{1}{3}\times \left\{\frac{3n}{2}\right(2a+(3n-1)d\left)\right\}=\frac{z}{3}$

$\Rightarrow \frac{z}{y-x}=3$