# Stephanie is planning to buy a big screen TV in 2 years. She believes the selling price will be \$1,800 at that time. Stephanie decides to make monthly deposits into an account that pays 4.8% /a, compounded monthly. How large does each deposit have to be so that she has enough money to buy the TV in 2 years?

Stephanie needs to deposit \$71.32 per month for two years to have the required amount of \$1,800 at the end of two years.

Stephanie plans to buy a big screen TV in 2 years. She believes the selling price will be \$1,800 at that time. Stephanie decides to make monthly deposits into an account that pays 4.8% per annum, compounded monthly.

Let the monthly deposits that Stephanie needs to make be P. The interest rate is 4.8% per annum, compounded monthly. The effective monthly rate is equal to 0.048/12 = 0.004. Her first deposit made into the account increases to P*(1+0.004)^24 after 2 years. The next deposit increases to P*(1+0.004)^23, and similarly, the final deposit is equal to P*(1+0.004)^1 at the end of 2 years.

The total amount she has at the end of 2 years is equal to P*(1+0.004)^24 + P*(1+0.004)^23 + ... P*(1+0.004)^1

P*(1+0.004)^24 + P*(1+0.004)^23 + ... P*(1+0.004)^1

= P((1+0.004)^24 + (1+0.004)^23 + ... (1+0.004)^1)

= P((1.004)^24 + (1.004)^23 + ... (1.004)^1)

= P*1.004*(1.004^24 - 1)/(1.004 - 1)

= P*25.237.

As this has to equal the cost of the TV, P*25.237 = 1,800.

P = 71.32.

Stephanie needs to deposit \$71.32 per month for two years to have the required amount of \$1,800 at the end of two years.

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