10) You have $1,000 to invest for retirement and you plan to add $500 every year to the account. You findan account that pays 9% interest per year. How much will you have in your retirement account by thetime that you retire in 54 years?a.$28,000.00b.$30,520.00C.$157,442.56d.$682,526.75A e.$1,732,448.30

Name: 10) You have $1,000 to invest for retirement and you plan to add $500
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Description: 10) You have $1,000 to invest for retirement and you plan to add $500 every year to the account. You findan account that pays 9% interest per year. How much will you have in your retirement account by thetime that you retire in 54 years?a.$28,000.00

Answered: 10) You have $1,000 to invest for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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10) You have $1,000 to invest for retirement and you plan to add $500 every year to the account. You find
an account that pays 9% interest per year. How much will you have in your retirement account by the
time that you retire in 54 years?
a.
$28,000.00
b.
$30,520.00
C.
$157,442.56
d.
$682,526.75
A e.
$1,732,448.30

Expert Solution

Step 1

We will use the following notations

P : Principal amount ,

i : rate of interest per year ,

n : number of years,

$A_{n}$ : accrued amount after n years .

After 1 year ,

$A_{1} = (P + P i) + 500 A_{1} = P (1 + i) + 500$ ( as we plan to add $ 500 every year )

Now A₁ will be the principal amount for the 2nd year ,

After the 2nd year ,

$A_{2} = (A_{1} + A_{1} i) + 500 A_{2} = A_{1} (1 + i) + 500$ ,

By using the above value of A₁ , we get

$A_{2} = [P (1 + i) + 500] (1 + i) + 500 A_{2} = P {(1 + i)}^{2} + 500 (1 + i) + 500 A_{2} = P {(1 + i)}^{2} + 500 [1 + (1 + i)]$

Now A₂ will be the principal amount for the 3rd year ,

After the 3rd year ,

$A_{3} = (A_{2} + A_{2} i) + 500 A_{3} = A_{2} (1 + i) + 500$ ,

By using the above value of A₂ , we get

$A_{3} = [P {(1 + i)}^{2} + 500 \times {1 + (1 + i)}] (1 + i) + 500 A_{3} = P {(1 + i)}^{3} + 500 \times {(1 + i) + {(1 + i)}^{2}} + 500 A_{3} = P {(1 + i)}^{3} + 500 [1 + (1 + i) + {(1 + i)}^{2}]$

Considering the above discussion , we can see a pattern as follows :

$A_{n} = P {(1 + i)}^{n} + 500 [1 + (1 + i) + {(1 + i)}^{2} + . . . . . . + {(1 + i)}^{n - 1}]$ ........(1)

We shall prove above formula is correct , using Mathematical induction .

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