For the Rule of 78, for a 12-month period, the last term in the sequence is 12 and the series sums to 78.

For an 10 month period, the last term is and the series sum is .

For a 15 month period, the last term is and the series sum is .

For a 20 month period, the last term is and the series sum is .

The nth term is n.

The sum of n terms is n(n+1)/2

Can you please show me how to solve. I appreciate it.

a. 55

To calculate the last term in the sequence and the series sum for the Rule of 78, we need to understand the formula and then apply it to each scenario.

The Rule of 78 is a method used to determine how interest charges are allocated over the life of a loan. It assumes that each month's interest payment is a fraction of the total interest charges, and this fraction decreases each month. The Rule of 78 formula can be expressed as:

Term = (Original term * (Original term + 1) / 2) - (N * (N - 1) / 2)
Sum = (Original term * (Original term + 1) * (2 * Original term + 1) / 6) - (N * (N - 1) * (2 * N - 1) / 6)

Where:
- Original term is the original length of the loan or repayment period.
- N is the number of completed months.

Now let's calculate the values for each scenario:

1. For a 10-month period:
- Original term: 12
- N: 10

Using the formula:
Term = (12 * (12 + 1) / 2) - (10 * (10 - 1) / 2)
Sum = (12 * (12 + 1) * (2 * 12 + 1) / 6) - (10 * (10 - 1) * (2 * 10 - 1) / 6)

2. For a 15-month period:
- Original term: 12
- N: 15

Apply the formula:
Term = (12 * (12 + 1) / 2) - (15 * (15 - 1) / 2)
Sum = (12 * (12 + 1) * (2 * 12 + 1) / 6) - (15 * (15 - 1) * (2 * 15 - 1) / 6)

3. For a 20-month period:
- Original term: 12
- N: 20

Using the formula:
Term = (12 * (12 + 1) / 2) - (20 * (20 - 1) / 2)
Sum = (12 * (12 + 1) * (2 * 12 + 1) / 6) - (20 * (20 - 1) * (2 * 20 - 1) / 6)

By substituting the values of Original term and N into the formulas, you can find the last term and the series sum for each scenario based on the Rule of 78.