A fast food restaurant offers to give away two 100-pound prizes on one day. They will give away four 100-pound prizes the next day, six 100-pound prizes the next day, and so on, giving away two more 100-pound prizes each day than the previous day. If 𝑛 represents the number of days in their campaign, find a formula to calculate how many pounds they will have given away in total by the end of the campaign.
Let’s look carefully at how much money the restaurant are giving away each day. On the first day, they’re giving away two 100-pound prizes, so that’s 200 pounds. The next day, they’re giving away four 100-pound prizes, so that’s 400 pounds. On the third day, they’re giving away six 100-pound prizes, so that’s 600 pounds. And then, this continues. Every day they give away two more 100-pound prizes, so 200 pound more than they gave away the day before.
We are asked to find a formula to calculate the total amount of money they give away if they run this campaign for 𝑛 days. Now, these terms, the amounts they give away every day — 200, 400, 600, 800, and so on — form an arithmetic sequence because they have a common difference of 200. We should recall that there is a formula for calculating the sum of the first 𝑛 terms of an arithmetic sequence. It’s 𝑠 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one 𝑑, where 𝑠 sub 𝑛 represents the sum of the first 𝑛 terms, 𝑎, or sometimes 𝑎 one, represents the first term, and 𝑑 represents the common difference.
We can now use this formula to help us find a formula for the total amount the restaurant will give away. We don’t know how many days they’re running the campaign for, so our formula will be in terms of the unknown 𝑛. The first term in the sequence is the amount the restaurant gives away on day one. That’s 200 pounds, or we’ll just use 200 in our formula. The common difference 𝑑 is the amount that the terms in the sequence increase by each time; that’s also 200 pounds. Substituting 200 for both 𝑎 and 𝑑 gives 𝑠 sub 𝑛 is equal to 𝑛 over two multiplied by two times 200 plus 200 multiplied by 𝑛 minus one. This simplifies to 𝑛 over two multiplied by 400 plus 200𝑛 minus 200. And then simplifying further within the parentheses, we have 𝑛 over two multiplied by 200 plus 200𝑛.
We can then cancel a factor of two throughout, which will give 𝑛 multiplied by 100 plus 100𝑛. And distributing the parentheses gives 100𝑛 plus 100𝑛 squared. This would be a perfectly acceptable form in which to leave our answer, but we’re going to choose to factor by 100. Doing so gives 100 multiplied by 𝑛 squared plus 𝑛. The restaurant can then use this formula to work out how much money they will give away in total depending on how many days they run the campaign for. For example, if they run the campaign for 20 days, they would substitute 𝑛 equals 20 to work out the total amount of money they give away. Our answer is 100 multiplied by 𝑛 squared plus 𝑛, where 𝑛 represents the number of days in the campaign.