The smiths spent 2,000 on clothes in a particular year if they increase the amount they spend on clothes by 10% each year the smiths will spend a total

They always increase their spending by 10%, so we can always take the previous year's spending and find 110% of it by multiplying by 1.10.

They start at 2,000

In 1 year, they spend 2000×1.1

In 2 years, they spend 2000×1.1×1.1

In 3 years, they spend 2000×1.1×1.1×1.1

In 4 years, they spend 2000×1.1×1.1×1.1×1.1

Because they only ask for four years, you could probably just multiply that out. But what if they asked for the 20th year? or the fiftieth? You wouldn't want to calculate that, you would want to find a pattern.

Well really, we want to keep track of repeated multiplication, which we can do by using exponents. Look carefully at the pattern:

year 1 = 2000×1.11

year 2 = 2000×1.12

year 3 = 2000×1.13

year 4 = 2000×1.14

Do you see it?

year n = 2000×1.1n

Finding patterns like this are going to be critical for geometric series