A company, which is making 200 mobile phones each week, plans to increase its production - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 1

To find the total number of mobile phones made during the first 52 weeks, we will calculate both the terms before week 21 and after.

1. For the first 21 weeks (weeks 1 to 21):

   • The production forms an arithmetic series where:
     • The first term, $a = 200$ (week 1 production)
     • The last term, $l = 600$ (week 21 production)
     • The number of terms, $n = 21$
   • The sum of an arithmetic series is given by: $S_n = \frac{n}{2}(a + l)$
   • Therefore: $S_{21} = \frac{21}{2}(200 + 600) = \frac{21}{2}(800) = 21 \times 400 = 8400$

2. For weeks 22 to 52:

   • Starting from week 22, the company produces 600 phones each week for 31 weeks:
   • The total production during these weeks is: $600 \times 31 = 18600$

3. Total production over 52 weeks:

   • Adding both sums together: $Total = S_{21} + S_{31} = 8400 + 18600 = 27000$

Thus, the total number of mobile phones made in the first 52 weeks is 27000.