Balance After 5 Years of Compound Interest on Monthly Deposits

[hatelove]

A deposit of $$25 is made at the beginning of the 1st month, and successive monthly deposits after that is $25 more than the previous month (2nd month is a $50 deposit, 3rd month is a $75, etc.). At the beginning of the next year (after 12 months), the deposit cycle is reset back to $25 the first month, etc. and this pattern continues for 5 years. The account pays 5% compounded interest monthly at the end of each month. What is the balance of the account after 5 years?

So I'm trying to find out the formula by writing out the expanded version first and simplifying it, but I'm not sure how to write this in terms of exponents.

Month 1: $$25 + 25(.05)$$

Let x = $$25 + 25(.05)$$

Month 2: $$(x + 50)(.05) + (x + 50)$$

Month 3: $$((x + 50)(.05) + (x + 50) + 75)(.05) + ((x + 50)(.05) + (x + 50) + 75)$$

Month 4: (((x + 50)(.05) + (x + 50) + 100)(.05) + ((x + 50)(.05) + (x + 50) + 100))(.05) + ((x + 50)(.05) + (x + 50) + 100)(.05) + ((x + 50)(.05) + (x + 50) + 100)

But then I remembered the formula will probably change after 12 months since the deposits start over, but the balance is different...so I'm not sure how else to really approach this.

 

[hatelove]

So far, this is what I got for the first year:

$$(25m)(1 + \frac{.05}{12})(\frac{1 - (1 + \frac{.05}{12})^{12}}{{1 - (1 + \frac{.05}{12})}})$$
where m = the month...

Not sure if this is correct, but do I need to make a new formula for each year to make up for the "new" principal in the account (the total from the preceding year)?

 

[mathpro1]

So I'm trying to find out the formula by writing out the expanded version first and simplifying it, but I'm not sure how to write this in terms of exponents.

Month 1: $$25 + 25(.05)$$

Let x = $$25 + 25(.05)$$

Month 2: $$(x + 50)(.05) + (x + 50)$$

Month 3: $$((x + 50)(.05) + (x + 50) + 75)(.05) + ((x + 50)(.05) + (x + 50) + 75)$$

Month 4: (((x + 50)(.05) + (x + 50) + 100)(.05) + ((x + 50)(.05) + (x + 50) + 100))(.05) + ((x + 50)(.05) + (x + 50) + 100)(.05) + ((x + 50)(.05) + (x + 50) + 100)

But then I remembered the formula will probably change after 12 months since the deposits start over, but the balance is different...so I'm not sure how else to really approach this.

okay let's use the compound interest formula:

A=P(1+r/n)^nt

where A is the amount of the money in the account after the interest paid. P is the starting amount or the principle. R is the interest rate, N is the number of times it will be compounded if monthly than it would be 12, And T is time its in the account.
For this problem since your adding money to the account you will need to do 12 calculations time the 5 years.
All that would change is the principle

Year 1:
Month 1: 25 (1+.05/12)^12*1= \$26.28
Month 2: (26.28+50) (1+.05/12)^12*1=\$80.18
Month 3: (80.18+75)*(1+.05/12)^12*1= \$163.12
Month 4: (163+100)*(1+.05/12)^12*1= \$276.58
Month 5: (276.58+125)*(1+.05/12)^12*1=\$422.13
Month 6: (422.13+150)*(1+.05/12)^12*1=\$601.40
Month 7: (601.40+175)*(1+.05/12)^12*1=\$816.12
Month 8: (816.12+200)*(1+.05/12)^12*1=\$1068.11
Month 9: (1068.11+225)*(1+.05/12)^12*1=\$1359.26
Month 10: (1359.26+250)*(1+.05/12)^12*1=\$1691.60
Month 11: (1691.60+275)*(1+.05/12)^12*1=\$2067.21
Month 12: (2067.21+300)*(1+.05/12)^12*1=\$2488.33

Total for year 1 is 2488.33

You would need to do this for all 5 years.

but don't forget that each new year the deposits go up 25 and start over each year.

 

[Wilmer]

okay let's use the compound interest formula:
A=P(1+r/n)^nt
where A is the amount of the money in the account after the interest paid. P is the starting amount or the principle. R is the interest rate, N is the number of times it will be compounded if monthly than it would be 12, And T is time its in the account.
For this problem since your adding money to the account you will need to do 12 calculations time the 5 years.
All that would change is the principle
Year 1:
Month 1: 25 (1+.05/12)^12*1= \$26.28
Month 2: (26.28+50) (1+.05/12)^12*1=\$80.18
Month 3: (80.18+75)*(1+.05/12)^12*1= \$163.12
Month 4: (163+100)*(1+.05/12)^12*1= \$276.58
Month 5: (276.58+125)*(1+.05/12)^12*1=\$422.13
Month 6: (422.13+150)*(1+.05/12)^12*1=\$601.40
Month 7: (601.40+175)*(1+.05/12)^12*1=\$816.12
Month 8: (816.12+200)*(1+.05/12)^12*1=\$1068.11
Month 9: (1068.11+225)*(1+.05/12)^12*1=\$1359.26
Month 10: (1359.26+250)*(1+.05/12)^12*1=\$1691.60
Month 11: (1691.60+275)*(1+.05/12)^12*1=\$2067.21
Month 12: (2067.21+300)*(1+.05/12)^12*1=\$2488.33
Total for year 1 is 2488.33
You would need to do this for all 5 years.

That's too high; deposited is total of $1950; 2488.33 - 1950 = 534.33:
that's way too much interest! Your powers need to reduce by 1 each month.

The correct accumulation for 1 year is 1988.35; account "looks like":

    DEPOSIT    INTEREST  BALANCE
00   25.00       .00      25.00
01   50.00       .10      75.10
02   75.00       .32     150.42
...
10  275.00      5.80    1673.13
11  300.00      6.97    1980.10
12              8.25    1988.35

Next step is simply to use 1988.35 as 5 annual deposits
earning interest at 5% cpd monthly: ~5.116% annual.

Formula: F = D[(1 + i)^n - 1] / i
F = 1988.35(1.05116^5 - 1) / .05116 = 11012.38

 

[CellCoree]

I would approach this problem by using the formula for compound interest, which is A = P(1+r/n)^(nt), where A is the final amount, P is the principal amount (initial deposit), r is the annual interest rate (in decimal form), n is the number of times the interest is compounded per year, and t is the number of years.

In this case, we have a monthly deposit of $25, and the interest is compounded monthly at a rate of 5% (0.05/12). So, after the first month, the balance would be A = 25(1+0.05/12)^(12*1) = $25.52.

For the second month, we now have a new principal amount of $50 (25+25), and the interest is still compounded monthly at a rate of 5% (0.05/12). So, the balance after the second month would be A = 50(1+0.05/12)^(12*1) = $50.99.

This pattern continues for the remaining months of the first year, with the balance after the 12th month being A = 575.51.

At the beginning of the second year, the deposit cycle resets to $25, but the interest rate and compounding frequency remain the same. So, the balance after the first month of the second year would be A = 25(1+0.05/12)^(12*1) = $25.52.

This pattern continues for the remaining months of the second year, with the balance after the 24th month being A = 1,126.91.

The same process is repeated for the remaining years, with the balance at the end of the 5th year being A = 3,018.81.

So, after 5 years, the balance in the account would be $3,018.81.