# Find the amount of an annuity if \frac{\$220}{month} is paid into it for a period of 20 years,...

## Question:

Find the amount of an annuity if {eq}\frac{\$220}{month} {/eq}is paid into it for a period of 20 years, earning interest at the rate of 8% year compounded continuously.

## Amount of Annuity:

{eq}\\ {/eq}

The continuously compounded Annuity can be obtained using the formula {eq}\boxed{A=P\dfrac{e^{rn}-1}{e^r-1}}
{/eq}, where **P** is the Payment made annually, **r** is the Rate of Interest per year and **n** is the time for which the payment is made.

An annuity is generally many numbers of installments that are made annually, monthly or maybe continuously.

## Answer and Explanation: 1

{eq}\\ {/eq}

Given : Payment paid per month= **220 $**.

So, the yearly payment will be {eq}220\times 12=2640 \ $ {/eq}

{eq}\Rightarrow P=2640 \ $. {/eq}

**n**= Number of years = **20 years.**

**r**= Rate of Interest = **8%** per year = **0.08** per year.

As we know that the formula for Annuity compounded continuously is given by:-

{eq}A=P\dfrac{e^{rn}-1}{e^r-1} {/eq}

Plugging the given values, we get :-

{eq}\Rightarrow A=2640\dfrac{e^{20\times 0.08}-1}{e^{0.08}-1} {/eq}

{eq}\Rightarrow A=2640\dfrac{e^{1.6}-1}{e^{0.08}-1} {/eq}

{eq}\Rightarrow A=2640\dfrac{4.96-1}{1.08-1} {/eq}

{eq}\Rightarrow A=2640\dfrac{3.96}{0.08} {/eq}

{eq}\Rightarrow A=2640\times 49.5 {/eq}

{eq}\Rightarrow \boxed{A=130680 \ $.} {/eq}

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Chapter 21 / Lesson 15An annuity is a type of savings account that pays back the investor in the future. Learn the formula used to calculate an annuity's value, and understand the importance of labeling specific numbers to calculate an output over time.

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