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An exponential function for the amount in the account after T years if the interest of 8% is compounding annually on initial amount 1700 is: P_T = 1700(1.08)^T
How to form mathematical expression from the given description?
You can represent the unknown amounts by the use of variables. Follow whatever the description is and convert it one by one mathematically. For example if it is asked to increase some item by 4 , then you can add 4 in that item to increase it by 4. If something is for example, doubled, then you can multiply that thing by 2 and so on methods can be used to convert description to mathematical expressions.
How does compound interest works?
Instead of calculating the interest on the initial amount, the compound interest repeats calculating interest on (old amount + interest to be paid).
For this case, we're specified that:
- Initial amount deposited = $1700
- Rate of interest = 8% annual = 0.08 (converted percent to fraction).
Let P be the inital amount, and r be the rate of interest compounding per unit of time (in decimal)
Let [tex]P_t[/tex] be the amount after t unit of time(same unit as the interest).
Then after 1 year, the new amount would be:
[tex]P_{t+1} = P_t + 100r \%\text{ of }P_t}\\P_{t+1}= P_t + P_t \times r\\P_{t+1} = P_t(1+r)[/tex]
Thus, we get:
[tex]P_t = P_{t-1}(1+r) = P_{t-2}(1+r)(1+r) = P_{t-2}(1+r)^2\\\\P_t = P_{t-t} (1+r)^t = P_0(1+r)^t[/tex]
where P₀ is the initial amount (as amount at t = 0 is initial amount)
Thus, we get:
[tex]P_t = P(1+r)^t[/tex]
Here, putting P = 1700, r = 0.08, and t = T, we get:
[tex]P_T = 1700(1.08)^T[/tex]
Thus, an exponential function for the amount in the account after T years if the interest of 8% is compounding annually on initial amount 1700 is: [tex]P_T = 1700(1.08)^T[/tex]
Learn more about compound interest here:
https://brainly.com/question/11897800
