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 18 month period, the last term is and the series sum is .
For a 24 month period, the last term is and the series sum is .
For a 30 month period, the last term is and the series sum is .
To find the last term and the series sum for different periods using the Rule of 78, we need to understand the formula and how it works.
The Rule of 78 is a method used to allocate precomputed interest on an installment loan over its payment period. It assumes that the interest for each period is a fixed proportion of the total interest charge, and that the principal is repaid uniformly over the loan period.
Let's break down the formula and use it to find the last term and series sum for the given periods:
For a 12-month period:
Using the Rule of 78, the sum of the series is given by the formula:
Sum = (n * (n + 1)) / 2
Substituting n = 12 into the formula, we can find the series sum:
Sum = (12 * (12 + 1)) / 2
Sum = (12 * 13) / 2
Sum = 156 / 2
Sum = 78
Therefore, the series sum for a 12-month period is 78.
Now, let's find the last term for a 12-month period:
The last term can be found using the formula:
Last Term = series sum - (n * (n - 1)) / 2
Substituting n = 12 and the series sum = 78 into the formula, we can find the last term:
Last Term = 78 - (12 * (12 - 1)) / 2
Last Term = 78 - (12 * 11) / 2
Last Term = 78 - (132 / 2)
Last Term = 78 - 66
Last Term = 12
Therefore, the last term for a 12-month period is 12.
Now, let's repeat this process for the other periods:
For an 18-month period:
Sum = (18 * (18 + 1)) / 2 = 171
Last Term = 171 - (18 * (18 - 1)) / 2 = 9
For a 24-month period:
Sum = (24 * (24 + 1)) / 2 = 300
Last Term = 300 - (24 * (24 - 1)) / 2 = 12
For a 30-month period:
Sum = (30 * (30 + 1)) / 2 = 465
Last Term = 465 - (30 * (30 - 1)) / 2 = 15
Therefore:
- For an 18-month period, the last term is 9 and the series sum is 171.
- For a 24-month period, the last term is 12 and the series sum is 300.
- For a 30-month period, the last term is 15 and the series sum is 465.
To find the last term in the sequence and the series sum for different periods using the Rule of 78, we need to understand the formula and how it applies.
The Rule of 78 is a method used in finance to allocate precomputed interest charges for installment loans. It assumes that interest is paid off faster in the early months of the loan, making the later months less expensive.
In the Rule of 78, the sum of the sequence is always 78, regardless of the length of the period. The last term, however, varies depending on the period length. The formula for finding the last term is:
Last Term = Sum of the sequence - (Sum of remaining terms)
Let's calculate the values for different periods:
For a 12-month period:
Last Term = 12 - (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11)
Last Term = 12 - 66
Last Term = 12
Series Sum = 78
For an 18-month period:
Last Term = 18 - (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17)
Last Term = 18 - 153
Last Term = -135
Series Sum = 78
It seems there's a mistake or misunderstanding in the calculation for the last term. Could you please provide the correct last term and series sum values for an 18-month period so that I can assist you further?
for a 18 month period, the last term is 18 and the sum is
1+2+3+...+18 ( ----> sum(n) = n(n+1)/2)
= 18(19)/2
= 171
Do the others the same way.
I recall this rule as something we taught way back in the 60's
Here is a challenge for you:
Phone any bank, mortgage company, credit unit or any money institution to see of they have heard of or if they still use the "Rule of 78".
If not, contact your school board and tell them to remove this topic from the syllabus of the course, and btw, tell them to get an updated textbook.