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Alice puts 3400 dollar on her bank account with interest rate 3,7%.

How much interest does she receive in the second year?

I answered: $3525.8\cdot1.037-3400\cdot1,037=130.5$, but this is the incorrect answer.

Can someone help me?

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Simple interest is given by the formula

$$I(t) = P_0rt$$

where $P_0$ is the initial principal, $100r\%$ is the annual interest rate, and $t$ is the time measured in years. The initial principal is $\$3400$ and the interest rate is

$$\frac{3.7\%}{100\%} = 0.037$$

Thus, the amount of interest earned in the second year is the difference between the amount earned after two years and the amount earned after one year.

\begin{align*} I(2) - I(1) & = \$3400(0.037)(2) - \$3400(0.037)(1)\\ & = \$3400(0.037)(2 - 1)\\ & = \$3400(0.037)(1)\\ & = \$125.80 \end{align*}

Observe that if we replaced $1$ by $n$ and $2$ by $n + 1$, we would obtain the same answer. Thus, provided no money is added to or removed from the account, the interest earned in any year would also be $\$125.80$.

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This is simple interest. If an amount $P$ is invested at an annual simple interest rate of $r$, then every year the interest earned is just $Pr$. This is true regardless of which year it is, because simple interest is always computed on the original principal.

In your case, the interest earned every year is $$\underbrace{3400}_{\textrm{principal }P}\cdot \underbrace{0.037}_{\textrm{rate }r} =\underbrace{125.80}_{\textrm{annual interest }Pr}$$

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3400•0.037=125.8

3400+125.8=3525.8

3525.8•0.037=130.45

That should help you.

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After one year there will be: $3400\cdot(1.037) = 3525.8$ dollars on the account. This means that she will recive $3400\cdot(0.037) = 125.8$ dollars in interest in the second year.

The interest from the second year she will recive the third year, and so on.

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