Math Competition Problems - Problem of the week - 2 (AMC8 Problem)

Updated: Oct 18, 2020

"On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working 20 days?"
- Source - AMC8 2015, Problem 9


Since this is problem #9 in AMC 8, this should be relatively easy to solve and you should be able to solve it in under a minute. But a quick look will tell you that the arithmetic might take quite a bit of time. So what do we do?

Let's try listing it out

Day No. of widget sold Running total

1 1 1

2 3 4

3 5 9

4 7 16

5 9 25

......... and so on

Now that we have listed it out we can see a pattern.

The amount of widgets she sold on Day 1 is 1^2

The total no. of widgets she sold till Day 2 is 2^2

The total no. of widgets she sold till Day 3 is 3^2

The total no. of widgets she sold till Day 4 is 4^2

So the number of widgets she sold till Day n can be summed up by the expression n^2

So the total no. of widgets Janabel sold after working 20 days is 20^2 = 400.

This type of problem solving technique is called finding a pattern. Now try the following question on your own using the same technique and post your answers in the comments section below

"Each member of a club shook hands with every other member who came for a meeting. There were a total of 45 handshakes. How many members were present at the meeting?"

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