Future value of annuity with change in interest rate

[panda125]

Hi, The problem i am having states:

John Doe established a saving account for her son's college education by making annual deposits of 6,000 dollars at the beginning of each of the six years to a saving account paying 8 percent interest. At the end of the sixth year, the account balance was transferred to a bank paying 10 percent interest, and annual deposit of 6000 dollar were made at the end of each year from the seventh through the tenth year. What was the account balance at the end of the tenth year?

I understand the first part, which i have a total of 6 periods with payments of 6000 dollars with 8 Percent interest and also i understand that the first part is an annuity due. I just don't understand what to do after you transfer the total amount to new bank paying 10 percent interest

My teacher has informed me the answer to the question is 97,445, but I don't understand how to get to this point would very much appreciate some help for this question.

 

[tkhunny]

You can ponder the formulas you memorized all you want. If this fails, which it most assuredly will, you MUST learn the basic principles. Just learn to write them out in the most fundamental way. This one goes like this:

i = 0.08
r = 1+i = 1.08
j = 0.10
s = 1+j = 1.10

\(\displaystyle 6000\cdot\left(r+r^{2}+r^{3}+r^{4}+r^{5}+r^{6}\right)\cdot s^{4} + 6000\cdot\left(1+s+s^{2}+s^{3}\right)\)

If you must, you can observe that this form is equivalent:

\(\displaystyle 6000\cdot\left(\frac{r-r^{7}}{1-r}\right)\cdot s^{4} + 6000\cdot\left(\frac{1-s^{4}}{1-s}\right)\)

Or, maybe a little simpler

\(\displaystyle 6000\cdot\left(\frac{r^{7}-r}{i}\right)\cdot s^{4} + 6000\cdot\left(\frac{s^{4}-1}{j}\right)\)

I would have to agree with your teacher's answer.

 

[panda125]

\(\displaystyle 6000\cdot\left(r+r^{2}+r^{3}+r^{4}+r^{5}+r^{6}\right)\cdot s^{4} + 6000\cdot\left(1+s+s^{2}+s^{3}\right)\)


Thank you for the help, but can you explain the why S^4 is being raised to the fourth power and also explain why a one is being added into the second part of the annuity

 

[Denis]

\(\displaystyle 6000\cdot\left(r+r^{2}+r^{3}+r^{4}+r^{5}+r^{6}\right)\cdot s^{4} + 6000\cdot\left(1+s+s^{2}+s^{3}\right)\)
Thank you for the help, but can you explain the why S^4 is being raised to the fourth power and also explain why a one is being added into the second part of the annuity

s^4 is used in order to pay interest @ 10% for 4 years (years 7 to 10) on the amount at end of year 6.
The "one" is simply s^0; refers to the last payment (end of year 10) which will not earn any interest
during the 10year period.

 

[tkhunny]

It takes a little courage to list the payments backwards so that the exponents are in a more familiar increasing order.

 

[Denis]

Interest @ 8%
 0   6000.00              6000.00
 1   6000.00    480.00   12480.00
 2   6000.00    998.40   19478.40
 3   6000.00   1558.27   27036.67
 4   6000.00   2162.93   35199.60
 5   6000.00   2815.97   44015.57
 6       .00   3521.25   47536.82
Now switch to 10%
 7   6000.00   4753.68   58290.50
 8   6000.00   5829.05   70119.55
 9   6000.00   7011.96   83131.51
10  6000.00   8313.15   97444.66

That's what "the account" will look like: follow that?