A boy saves some money over a period of 60 weeks - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 2

The total savings over 60 weeks can be calculated using the formula for the sum of an arithmetic series:

$S_n = \frac{n}{2} \times (a + l)$

where:

• $n = 60$ (the total number of terms)
• $a = 10$ (the first term)
• $l = T_{60} = a + (n-1)d = 10 + (60 - 1) \times 5 = 10 + 295 = 305$ (the last term)

Substituting these into the formula:

$S_{60} = \frac{60}{2} \times (10 + 305) = 30 \times 315 = 9450$

Thus, the total amount saved over 60 weeks is £94.50.

To show that m(m + 1) = 35 × 36, we first calculate the right side:

$35 \times 36 = 1260$

Given that the sister saves money in an arithmetic sequence:

• The first term $a = 10$, and the common difference $d = 10$.
• The sum of her savings is expressed as:

$S_m = \frac{m}{2} \times (2a + (m-1)d)$

Substituting the values, we have:

$S_m = \frac{m}{2} \times (20 + 10(m-1)) = \frac{m}{2} \times (10m + 10) = 5m(m+1)$

Setting this equal to 63 pounds:

$5m(m+1) = 1260$

Thus, we simplify:

$m(m + 1) = 35 \times 36$

From the equation $m(m + 1) = 1260$, we can factor the left-hand side:

Finding integer solutions:

1. $m = 35$: In this case, $m + 1 = 36$

Thus, we find the value of m is:

$m = 35$