John is a collector of paintings. The first painting he bought costs £4500. Each subsequent painting costs £200 more than the previous one. Now, John owns 18 paintings. How much has he spent in total on paintings?

Instead of working out cost of each painting up to the 18th. Is there another way to do this? Question is from an online learning maths platform. Thank you in advance for your help.

The total cost is given by 4500 + an extra amount for each painting:

(4500 + 0) + (4500 + 200) + (4500 + 400) ..... + (4500 + 3200) + (4500 + 3400)

You can pair the extra amounts together, 1st with 18th, 2nd with 17th etc to give 9 * 3400.

so total cost = (4500 * 18) + (9 * 3400)

This is not my area of expertise. But would this work?

The first painting costs 4,500. The 18th painting costs 4,500+(17x200) because each of the 17 next paintings is 200 more than the previous one. So that's 7,900.

The average painting price is therefore 6,200 (4,500+7,900 then divided by 2).

The total amouont spent on paintings will be the number of paintings times the average (18x6,200) which is 111,600.

Thank you very much for taking the time to reply.