You opened a bank account at the beginning of the month and have added the same amount of money each day. On the 5th day, you have $80. On the 21st day, you have $160. How much did you originally open the account with? How much are you putting in the account each day? How much will you have on the 30th day of the month?

We are given:

Money on the fifth day (a₅) = $80

Money on day 21 (a₂₁) = $160

To Find:

Money in the bank on the first day (a₁) = ?

Money being added to the account per day (d) = ?

Money in the bank on day 30 (a₃₀) = ?

Finding the money on day one and the money added:

Solving for equations

Since we are adding a constant amount to our account every day, the amount of money in the bank can be represented by an AP

We know that the formula for the nth term of an AP is:

aₙ = a₁ + (n-1)d

So, we can write the amount of money on day 5 as:

a₅ = a₁ + (5-1)d

a₅ = a₁ + 4d

80 = a₁+  4d                        [We are given that a₅ = 80]

We are also given the amount of money on day 21:

a₂₁ = $160

Money on day 21 can be rewritten as:

a₂₁ = a₁ + (21-1)d

160 = a₁ + 20d                  [Since a₂₁ = 160]

Solving the equations:

Now we have 2 equations. We will now solve them to get the values of a₁ and d

160 = a₁ + 20d (1)

80 = a₁ + 4d(2)

From equation (2), we can say that:

a₁ = 80 - 4d              [subtracting 4d from both sides]

Now, we will use this value of a₁ in the equation (1):

160 = a₁ + 20d

160 = (80-4d) + 20d

160 = 80 + 16d

80 = 16d             [subtracting 80 from both sides]

d = 5                   [dividing both sides by 16]

Now that we know the value of d, we can use it in any of the 2 equations:

80 = a₁ + 4d

80 = a₁ + 4(5)              [since d=5]

80 = a₁ + 20

a₁ = 60                        [subtracting 20 from both sides]

Now, we know the values of the initial amount of money and the money added to the account everyday

Initial money in the bank = $60

Money added everyday = $5

Money in day 30:

We will use the AP formula to find the value of money on day 30

a₃₀ = a₁ + (30-1)d

We know that a₁ = $60 , d = $5

a₃₀ = 60 + 29(5)

a₃₀ = 60 + 145

a₃₀ = $205

Money on day 30 = $205

1/800 is the right answer