Bill earns 1 dime the first day, 2 dimes the second day, 3 dimes the third day, 4 dimes the fourth day, etc. How many days will it take him to earn $2.

On day 4 he has $1.00

Day 5: 1.00 + 0.50 = 1.50
Day 6: 1.50 + 0.60 = 2.10

To find out how many days it will take Bill to earn $2, we can break down the problem into smaller steps.

Step 1: Determine how much Bill earns each day.
From the given information, we can see that Bill earns 1 dime on the first day, 2 dimes on the second day, 3 dimes on the third day, and so on. This is an arithmetic sequence, where the nth term is given by n dimes.

Step 2: Calculate the sum of the earnings.
We want to find the total amount Bill earns over a certain number of days. Using the formula for the sum of an arithmetic series, we can calculate this. The formula is:

Sum = (n/2) * (first term + last term),

where n is the number of terms and the first and last terms are 1 dime and the last term, respectively.

Step 3: Set up the equation to solve for the number of days.
We want to find the number of days it takes for Bill to earn $2, which is equivalent to 200 dimes. So, we set up the equation:

200 = (n/2) * (1 + last term).

Step 4: Find the value of the last term.
To find the value of the last term, we need to determine the pattern followed by Bill's earnings. We can see that Bill earns 1 dime, 2 dimes, 3 dimes, and so on. This forms a linear relationship, where the last term is given by the number of days he has earned. Therefore, the last term is n dimes.

Step 5: Simplify and solve the equation.
Substituting the value of the last term as n, we get:

200 = (n/2) * (1 + n).
Multiplying both sides by 2 to eliminate the fraction:

400 = n * (1 + n).

Step 6: Solve the quadratic equation.
Rearranging the equation, we get:

n^2 + n - 400 = 0.

Using the quadratic formula:

n = (-b ± √(b^2 - 4ac)) / (2a),

with a = 1, b = 1, and c = -400, we can solve for n.

Step 7: Determine the positive integer solution.
When we solve the quadratic equation, we get two solutions: n ≈ 19.45 and n ≈ -20.45. Since the number of days cannot be negative, we consider the positive solution, which is approximately 19.45.

Therefore, it will take Bill approximately 19 days to earn $2.