Answer:
The constant percent increase per year is 8%.
[tex]B(T) = 2000(1.08)^{T}{/tex]
Step-by-step explanation:
The amount in the balance after T years is given by the following equation:
[tex]B(T) = B(0)(1+r)^{T}[/tex]
In which B(0) is the initial amount and r is the growth rate, as a decimal.
Next year’s balance = 1.08 x current balance
So
[tex]B(1) = (1.08)B(0)[/tex]
This means that:
[tex]1 + r = 1.08[/tex]
[tex]r = 1.08 - 1[/tex]
[tex]r = 0.08[/tex]
The constant percent increase per year is 8%
If the original balance of the investment was $2000 find a formula that gives the balance B (in dollars) after T years.
This means that [tex]B(0) = 2000[/tex]
So
[tex]B(T) = B(0)(1+r)^{T}[/tex]
[tex]B(T) = 2000(1+0.08)^{T}[/tex]
[tex]B(T) = 2000(1.08)^{T}{/tex]
Answer:
[tex]\large \boxed{1. 8 \, \%; \, 2. \, B_{T} = 2000(1.08)^{T}}[/tex]
Step-by-step explanation:
1. Rate of increase
Let B = this year's balance. Then
1.08B = next year's balance
1.08B = B + 0.08B = B + 8 % of B
[tex]\text{The rate of increase of B is $\large \boxed{\mathbf{8 \%}}$ per year.}[/tex]
2. The formula
Each year, the balance is multiplied by 1.08.
The end of a year is the same as the end of the previous year.
If T = the number of years, then the balance at the beginning of Year 1 is B₀.
[tex]B_{T} = B_{0}(1.08)^{T}[/tex]
If B₀ = $2000, the formula becomes
[tex]\large \boxed{\mathbf{B_{T} = 2000(1.08)^{T}}}$[/tex]
