I need help in my homework question... I got a homework,I solved 6 que. But I could not solve 2 que. I tried but I could not... I need help..


que.1. Annuities
Calculate the present value of an annuity where $1000 is added each year for 9 years and
the given interest rate is 6%. Show your result as a partial geometric series and then as a
difference of two geometric series.

que.2 Hyperbolic Sine and Cosine Functions
For sinh(x) = (ex-e-x)/2 and cosh(x) = (ex+e-x)/2, prove the following identities:
a) cosh(x) = cosh(-x)
b) sinh(x) = -sinh(-x)
c) sinh(2x) = 2sinh(x)cosh(x)
d) cosh(2x) = cosh2(x) + sinh2(x)
e) cosh2(x) - sinh2(x)=1

I hope someone will help me soon

que. 2 is very straightforward; just substitute into the equations provided. For example:

a) cosh(-x) = (e^(-x) + e^(x))/2 = cosh(x)

c) 2*sinh(x)*cosh(x) = 2*(e^x - e^-x)/2*(e^x + e^-x)/2 = (e^2x - e^ -2x)/2 = sinh(2x)

e) cosh^2(x) - sinh^2(x) = (cosh(x) + sinh(x)) * (cosh(x) - sinh(x)) = (e^x) * (e^-x) = 1

Now can you do b and d?

Thank you so much... now I can do... do you know que.1?

Since you've invested $1000 per year for 9 years, you can calculate the total by adding up the current value of each year's contribution.

The amount invested 9 years ago is now worth 1000*(1 + 0.06)^9 = $1689.48
The amount invested 8 years ago is now worth 1000*(1 + 0.06)^8 = $1593.85
...
The amount invested 1 year ago is now worth 1000*(1 + 0.06)^1 = $1060

The total worth is the sum of all those contributions. (I'll leave it up to you to do the actual numbers).

So, we can write it as a partial geometric series:
V = 1000 * sum((1.06)^n, n=1:9)

This is a partial geometric series because the exponent, n, only goes from 1 to 9. In a full geometric series it would go from 1 to infinity.

This partial geometric series can be written as the difference of two full geometric series:

1000*sum(1.06^n,n=1:9) = 1000*sum(1.06^n, n=1:inf) - 1000*sum(1.06^n,n=10:inf)

By the way, when you add up the numbers from the first part, the answer should come out to be the same as the "shortcut" formula:

V = A * ((1 + r)^(n+1) - 1) / r - A,

where A is the amount you put in each year (1000), r is the interest rate (0.06), and n is the number of periods (9).

V = 1000*(1.06^10-1)/0.06 - 1000 = $12,180.79

Thank you so much :)