Noah started his action figure collection, where each year he buys eight action figures. Write an algebraic expression that can be used to find the number of action figures he would have after 𝑛 years and then determine how many action figures he would have after 24 years.
There are lots of ways of approaching this problem. As Noah buys the same number of action figures each year, we could calculate the total number of action figures by multiplying the number of action figures per year by the number of years. We are told that the number of action figures he buys each year is eight. And we’re asked to find an expression for the number of action figures he would have after 𝑛 years. This means that the total number of action figures would be eight multiplied by 𝑛 or eight 𝑛.
We are also asked to work out how many action figures Noah has after 24 years. This means that 𝑛 is equal to 24. We therefore need to multiply eight by 24. We know that eight multiplied by 20 is 160. Eight multiplied by four is equal to 32. The sum of these values will give us eight multiplied by 24, which is equal to 192. After 24 years, Noah will have 192 action figures.
An alternative method would be to use our knowledge of arithmetic sequences. At the end of the first year, Noah has eight action figures. After the second year, he has 16. After the third year, he has 24, and so on. The general term of any arithmetic sequence written 𝑎 sub 𝑛 is equal to 𝑎 sub one plus 𝑛 minus one multiplied by 𝑑, where 𝑎 sub one is the first term of the sequence and 𝑑 is the common difference.
In this question, both 𝑎 and 𝑑 are equal to eight. Therefore, 𝑎 sub 𝑛 is equal to eight plus 𝑛 minus one multiplied by eight. By distributing our parentheses, this simplifies to eight plus eight 𝑛 minus eight. As eight minus eight is equal to zero, 𝑎 sub 𝑛 is equal to eight 𝑛. This confirms that the number of action figures Noah would have after 𝑛 years is eight 𝑛.
We can once again substitute 𝑛 equals 24 into this expression to show that after 24 years Noah would have 192 action figures.