Explain why the amount of money in the account at the end of t years is given by the formula: p = p0(1 + r/n)^nt

See  below.

Step-by-step explanation:

This is the formula for the value of an investment with compound interest.

PO is the initial amount invested, P is the amount after investing t years , r is  yearly rate of interest (as a fraction) and n is the number of times interest is paid per year.

An example using values will serve to explain it further.  If the rate is 2% ( which = 0.02) per year and  the interest is added to the account every 6 months ( twice a year)  then, in the parentheses we have ( 1 + 0.02/2). The 1 is there because we are talking about the amount of money PLUS the interest. Now if we are dealing with 5 year's investment , the exponent nt will be 2*5 = 10 because  the interest is paid 2 times a year over 5 years: - 10 payments. So if we invest 10,000 at 2% interest over 5 years with 2 interest payments a year the total after 5 years  = P = 10,000(1 + 0.02/2)^10 = 11,046.22.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!