Sue is training for a marathon - Edexcel - A-Level Maths Pure - Question 9 - 2008 - Paper 1

The length of her training run on the nth Saturday can be modeled by the formula:

$t_n = 5 + 2(n - 1)$

This expression accounts for the initial 5 km run, with an increase of 2 km for each subsequent Saturday.

To find the total distance she runs, we calculate the distance for each Saturday:

The n-th term, where each Saturday's distance is:

S_n = rac{n}{2} imes (2a + (n - 1)d)

where $a = 5$ km and $d = 2$ km:

Substituting the values:

S_n = rac{n}{2} imes (2 imes 5 + (n - 1) imes 2)

This simplifies to:

S_n = rac{n}{2} imes (10 + 2n - 2) = rac{n}{2} imes (2n + 8) = n(n + 4)

On the nth Saturday, Sue runs 43 km, therefore:

$43 = 5 + 2(n - 1)$

Solving for n:

$43 - 5 = 2(n - 1)$
$38 = 2(n - 1)$
$38/2 = n - 1$
$19 = n - 1$
$n = 20$

Using the established formula for total distance:

When $n = 20$, we substitute into:

$S_n = n(n + 4) = 20(20 + 4) = 20 imes 24 = 480 ext{ km}$

Thus, in n weeks of training, Sue runs a total distance of 480 km.