Question

Solve Word Problem Applying Arithmetic Sequence:

On the first day of each month, Ms. Aida deposits a fixed amount in a mutual fund. On June 1, her account balance was Php 1,000, and on December 1, her account balance will be Php 4,000. How much does she have deposited each month?​

Answer 1

Answer:

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.

We can find the value of the annuity right after the last deposit by using a geometric series with \displaystyle {a}_{1}=50a

1

=50 and \(r=100.5%=1.005\). After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.

We can find the value of the annuity after \displaystyle nn deposits using the formula for the sum of the first \displaystyle nn terms of a geometric series. In 6 years, there are 72 months, so \displaystyle n=72n=72. We can substitute \displaystyle {a}_{1}=50, r=1.005, \text{and} n=72a

1

=50,r=1.005,andn=72 into the formula, and simplify to find the value of the annuity after 6 years.

\displaystyle {S}_{72}=\frac{50\left(1-{1.005}^{72}\right)}{1 - 1.005}\approx 4\text{,}320.44S

72

=

1−1.005

50(1−1.005

72

)

≈4,320.44

After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of \(72\left(50\right) = $3,600\). This means that because of the annuity, the couple earned $720.44 interest in their college fund.

Answer 2

Given:

Account balance of Aida in June = 1000

Account balance of Aida in December = 4000

To find:

Amount Aida deposits each month

Solution:

To solve in the form of an AP, let a₆ = 1000 and a₁₂ = 4000. We need to determine common difference, $d$ which gives the amount Aida deposits on first day of each month.

$a_{n}=a+(n-1)d$

$\\ a_{6}=a+(6-1)d$

$\\ 1000 =a+5d...(1)$

$a_{12}=a+(12-1)d$

$4000=a+11d...(2)$

Subtracting equation (1) from equation (2)

$3000=6d$

$d=\frac{3000}{6}$

∴ $d=500$

Aida deposits Php 500 on the first day of each month.