A loan of $5600 is to be repaid at 9% compounded annually by making 10 payments at the end of each quarter. Each of the last six payments are two times the amount of each of the first four payments. What is the size of each payment?

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A loan of $5600 is to be repaid at 9% compounded annually by making 10 payments at the end of each quarter. Each of the last six payments are two times the amount of each of the first four payments. What is the size of each payment?

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You should abandon your trouble and practice BASIC PRINCIPLES. You will never struggle with the "right formula" again.

A loan of $5600 is to be repaid at 9% compounded annually by making 10 payments at the end of each quarter. Each of the last six payments are two times the amount of each of the first four payments. What is the size of each payment?

i = 0.09

j = (1+i)^(1/4) - 1

v = 1/(1+j)

That's all you need. Now build it.

5600 = Pmt(v + v^2 + v^3 + v^4 + 2(v^5 + v^6 + v^7 + v^8 + v^9 + v^10))

This leads directly to:

\(\displaystyle 5600 = Pmt\cdot\left[\left(\frac{v-v^{5}}{1-v}\right) + 2\cdot\left(\frac{v^{5} - v^{11}}{1-v}\right)\right] = \frac{Pmt\cdot v}{1-v}\cdot\left[\left(1-v^{4}\right)+2\cdot\left(v^{4}-v^{10}\right)\right]\)

Simplify a little more and GO!

1) Okay, so I get i = 0.021778181WHAT trouble are you having?

You need to do these steps ( let p = payment at start) :

1: convert 9% annual to its equivalent quarterly rate:

(1 + i)^4 = 1.09 ; solve for i which will be the quarterly rate.

2: get the future value of 5600: f = 5600(1 + i)^10

3: get future value of first 4 payments:

u = p[(1 + i)^6][(1 + i)^4 - 1] / i

4: get future value of last 6 payments: v = 2p[(1 + i)^6 - 1] / i

So we have: f = u + v : solve for p

2) 6946.31

3) [5600(1.137993409)(.009)] / i = 26335.93

4) [11500(.137993409)] / i = 70966.73

Have I done it correctly so far? And since there's no specified dates for these payments, would I simply divide #3 by 4 and #4 by 6?

Edit: Oh, and I'm not sure where p comes from when you ask to solve for it at the end

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This might be more enlightening5600 = Pmt(v + v^2 + v^3 + v^4 + 2(v^5 + v^6 + v^7 + v^8 + v^9 + v^10))

5600 = Pmt*(v + v^2 + v^3 + v^4 + v^5 + v^6 + v^7 + v^8 + v^9 + v^10) + Pmt*(v^5 + v^6 + v^7 + v^8 + v^9 + v^10)

5600 = Pmt*v*(1 + v + v^2 + v^3 + v^4 + v^5 + v^6 + v^7 + v^8 + v^9) + Pmt*(v^5)*(1 + v + v^2 + v^3 + v^4 + v^5)

This second one is very instructive. Look how it rewords the description. Rather than increasing the payment, it suggests a level payment stream for the entire period, increased by a different payment stream deferred five periods!

5600 = Pmt*v*[(1 + v + v^2 + v^3 + v^4 + v^5 + v^6 + v^7 + v^8 + v^9) + (v^4)*(1 + v + v^2 + v^3 + v^4 + v^5)]

\(\displaystyle 5600 = Pmt\cdot v\cdot\left[\left(\frac{1-v^{10}}{1-v}\right) + v^{4}\cdot \left(\frac{1-v^{6}}{1-v}\right)\right]\)

\(\displaystyle 5600 = Pmt\cdot \frac{v}{1-v}\cdot\left[\left(1-v^{10}\right) + v^{4}\cdot \left(1-v^{6}\right)\right]\)

Really, a little algebra and the world is a better place. Unique answers don't care how you find them.

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