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If a bank account pays 5% yearly interest, that means if you have \$$x$ in the account, then after a year the bank will add 5% of \$$x$ to the account. Since 5% of \$$x$ is the same as \$$0.05x$, if you start with \$$x$, a year later you will have $\$x+\$0.05x=\$1.05x$. So every year, the amount of money in the account is multiplied by 1.05.
In general:
If a bank account pays $r$% yearly interest, then after $n$ years the amount of money in your account will be multiplied by $$(1+r/100)^n$$.
If a credit card charges 2% monthly interest, that means if you owe \$$x$, then after a month the credit card company will charge you 2% of \$$x$. You will then owe a total of $\$x+\$0.02x=\$1.02x$. So each month, the amount of money you owe is multiplied by 1.02.
If your debt is being multiplied by $$1+r/100$$ every year, you are being charged $r$% yearly interest. For example, if it was being multiplied by $1.15$ every year, then that would mean you were being charged 15% yearly interest.
A Mathscribe Munchies vending machine costs \$1000. Every year you own it, you will make \$200 in profit. So, if you own one, the total profit you will make after $t$ years is:
This function $P(t)$ is graphed on the grid to the left. (When your total profit is negative, that means that you still have less money than when you started.)
The Mathscribe Munchies company decides to offer a \$200 rebate when you buy a vending machine from them. This means that, if you buy the machine, you’ll have an extra \$200 to start with, so your total profit after $t$ years will be $P(t)+200$. Click to see a graph of $P(t)+200$ along with the graph of $P(t)$.
You now would like to think about buying three Mathscribe Munchies machines instead of one. If you do this, your profit after $t$ years will be $3P(t)$. This is graphed on the grid to the left, along with the graph of $\cl"red"{P(t)}$. (The Mathscribe Munchies company no longer offers rebates on its vending machines.)
There is a story that’s at least a thousand years old, about a wise man who served his king well for many years. When the king asked him what he wanted as a reward for his service, he said:
“On the first square of this chessboard, I would like you to put a grain of rice. On the next square, put two grains of rice. On the square after that, put four grains of rice. On each square, put twice as many grains of rice as you did on the square before. Once you have done this for every square on the chessboard, I will take that rice as my reward.”
We’ll look at the sequence $C(n)$ whose first few terms are
$C(n)$ is the number of grains of rice that the king is supposed to put on the $n$th square of the chessboard.
Because $C(n)$ is a geometric sequence with common ratio $2$, there must be a formula for $C(n)$ which looks like $C(n)=a ⋅ 2^n$ for some number $a$.
This formula tells us that $C(1)=a ⋅ 2^1 = 2a$. Also, there should be one grain of rice on the first square, so $C(1)=1$. If $C(1)=2a$ and $C(1)=1$, then $2a=1$.