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If I invest amount A today for N years and then invest amount B every year for the same N years (as an ordinary annuity), then to find the future value at the end of N years should I treat the first investment as a single lump sum investment, calculate its future value, and then calculate the future value of the annuity separately?

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    Why not treat this as two completely separate problems and add the two values? – RonJohn Nov 18 '19 at 14:23
  • That's what I was thinking I needed to do; find the two future values (one from the first sum and then the annuity's) and combine them. This is what you're saying, yes? – user91685 Nov 18 '19 at 14:34
  • "This is what you're saying, yes?" Yup. – RonJohn Nov 18 '19 at 14:40
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I use GNU Octave for problems like these:

octave:1> A=1000
A =  1000
octave:2> N=10
N =  10
octave:3> B=200
B =  200
octave:4> k=1.08
k =  1.0800
octave:5> A*k^N + sum(B*k.^[1:N])
ans =  5288.0

Here I invested 1000 EUR for 10 years at rate of 8% = 0.08, and 200 EUR every year for 10 years at the same rate. At this rate, it will be 5288.0 EUR. (To find the missing decimal, use format long to get a more accurate answer.)

So, to answer your question:

should I treat the first investment as a single lump sum investment, calculate its future value, and then calculate the future value of the annuity separately?

You see that the formula has indeed the future values of both separately as an addition. So, the answer to this question is yes.

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