Artificial intelligent assistant

cash flow diagram, in/outflow series I have a econ midterm coming up soon and stumbled upon this question. I know this is the math section but it appears not many use the finance one. My approach is: $2C=\frac{800}{1.12^2}+\frac{1200}{1.12^6}=1245.71$ or $C=1245.71/2=622.85$ But I have a gut feeling this is wrong. I believe the answer is somewhere around $\$781$. **Consider the following cash flow diagram. What value of C makes the inflow series equivalent to the outflow series at an interest rate of 12% compounded annually?** !enter image description here

I would approach this question by observing the present value of all cash flows (in and out) at every point in time. Lets first begin with the outflows then inflows.

$$PV_{\mathrm{Outflow}}=2C + \sum_{i=2}^{8}\frac{C}{(1+12\%)^i}.$$

$$PV_{\mathrm{Inflow}}=\sum_{i=1}^{4}\frac{800}{(1+12\%)^i}+\sum_{i=5}^{8}\frac{1200}{(1+12\%)^i}$$

Now: " **What value of C makes the inflow series equivalent to the outflow series** ". Thus the equation I am trying to solve is

$$PV_{\mathrm{Inflow}} = PV_{\mathrm{Outflow}},$$

implying that

$$ \sum_{i=1}^{4}\frac{800}{(1+12\%)^i}+\sum_{i=5}^{8}\frac{1200}{(1+12\%)^i} = 2C + \sum_{i=2}^{8}\frac{C}{(1+12\%)^i}.$$

Furthermore

$$C = \frac{\sum_{i=1}^{4}\frac{800}{(1+12\%)^i}+\sum_{i=5}^{8}\frac{1200}{(1+12\%)^i}}{2 + \sum_{i=2}^{8}\frac{1}{(1+12\%)^i}}.$$

$$C = \$781.30\ (\mathrm{2dp.})$$

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