Question

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 14 month period, the last term is
and the series sum is
.

For a 16 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
.

Answer 1

The Rule of 78, also known as the sum-of-the-digits method, is used in calculating the prepayment penalties for loans. In this method, the sum of digits of the total number of payments (months, in this case) is used to calculate the interest on a loan.

To determine the last term and the series sum for different periods, you can calculate as follows:

1. **For a 12-month period:**
- The last term in the sequence: **12**
- The series sums to: **78**

2. **For a 14-month period:**
- The last term in the sequence is calculated as: \( \text{last term} = 14 \)
- The series sum is calculated as:
\[
\text{Series sum} = 14 + 13 + 12 + ... + 1 = \frac{14 \times (14 + 1)}{2} = \frac{14 \times 15}{2} = 105
\]

3. **For a 16-month period:**
- The last term in the sequence: **16**
- The series sum is calculated as:
\[
\text{Series sum} = 16 + 15 + 14 + ... + 1 = \frac{16 \times (16 + 1)}{2} = \frac{16 \times 17}{2} = 136
\]

4. **For a 20-month period:**
- The last term in the sequence: **20**
- The series sum is calculated as:
\[
\text{Series sum} = 20 + 19 + 18 + ... + 1 = \frac{20 \times (20 + 1)}{2} = \frac{20 \times 21}{2} = 210
\]

So the final results are:

- For a **14-month period**: Last term is **14**, Series sum is **105**.
- For a **16-month period**: Last term is **16**, Series sum is **136**.
- For a **20-month period**: Last term is **20**, Series sum is **210**.