If you deposit 100 at the end of every month into an...

If you deposit $$\$ 100$$ at the end of every month into an account that pays $3 \%$ interest per year compounded monthly, the amount of interest accumulated after $n$ months is given by the sequence $$ I_n=100\left(\frac{1.0025^n-1}{0.0025}-n\right) $$ (a) Find the first six terms of the sequence. (b) How much interest will you have earned after two years?

If you deposit $$\$ 100$$ at the end of every month into an account that pays $3 \%$ interest per year compounded monthly, the amount of interest accumulated after $n$ months is given by the sequence

$$
I_n=100\left(\frac{1.0025^n-1}{0.0025}-n\right)
$$

(a) Find the first six terms of the sequence.
(b) How much interest will you have earned after two years?

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Key Concepts

Interest Rate Conversion

Interest rate conversion involves adjusting an annual interest rate to match the compounding period in use. For instance, converting an annual rate to a monthly rate by dividing the annual rate by the number of periods per year is essential when calculations are based on monthly compounding.

Future Value of an Annuity

The future value of an annuity calculates the total accumulated value of a series of equal deposits made at regular intervals, after interest has compounded over a specified period. This concept is crucial in determining how much money will be available in the future, considering both the contributions and the compounded interest.

Compound Interest

Compound interest is the process by which interest earned on an investment is reinvested to earn additional interest, so that interest accumulates on both the original principal and the accumulated interest over time. This concept is fundamental in understanding how investments grow exponentially, especially when interest is compounded frequently, such as monthly.

Annuity

An annuity is a financial product or concept involving a series of equal payments made at regular intervals. In this context, it relates to making regular deposits into an account and analyzing how these periodic contributions grow with compound interest over time.

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