Question

Suppose Mrs. Reyes would like to save Php 1,000.00 at the end of each month for 9 months in a fund that gives 5% per annum compounded monthly. How much would the value of her savings be after 7 months?

Answer 1

The rate is r = 5% = 0.05
Compounding interval, n = 12, monthly compounding
Therefore
r/n = 0.05/12 = 0.004167

The first deposit has a duration of 7 months. Its value is
a₁ = 1000*(1.004167)⁷

The second deposit has a duration of 6 months. Its value is
a₂ = 1000*(1.004167)⁶
and so on.

The values after each month from a geometric sequence with
a = 1000*(1.004167)
r = 1.004167

Over 7 months, the total sum is
$\frac{1000*1.004167*(1-1.004167^{7} )}{1-1.004167} =7117.64$


Answer: Php 7,117.64

Answer 2

After 7 months her savings will be 7117.64.

Step-by-step explanation:

Given :

Rate is r = 5% = 0.05

Compounding interval, n = 12

Solution :

Now,

$\rm \dfrac{r}{n} = \dfarc{0.05}{12}$

$\rm \dfrac{r}{n} = 0.004167$

Now first deposit duration is 7 months. Therefore, its value is  $\rm a_1 = 1000\times (1.004167)^7$

Now second deposit duration is 6 months. Therefore, its value is

$\rm a_2 = 1000\times (1.004167)^6$

Now third deposit duration is 5 months. Therefore, its value is

$\rm a_3 = 1000\times (1.004167)^5$

and so on.

The values are in geometric sequence where

$\rm a = 1000\times (1.004167)$

$\rm r = 1.004167$

We know that the sum of the geometric sequence is given by,

$\rm S_n = \dfrac{a (r^n-1)}{(r-1)}$    ------ (1)

Now put the values of a and r in equation (1),

$\rm S_7=\dfrac{1000\times(1.004167)((1.004167)^7-1)}{(1.004167-1)}$

$\rm S_7 = 7117.64$

After 7 months her savings will be 7117.64.

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