Answer:
1. $990
2. Sn = Sn-1 + (r/100) * Sn-1
3. $27,037.44
Explanation:
1. The interest earned each year can be calculated using the simple interest formula:
Simple Interest = (Principal * Rate * Time) / 100
Here, Principal = $22,000, Rate = 4.5%, and Time = 1 year
So, the interest earned each year would be:
= (22,000 * 4.5 * 1) / 100
= $990
Therefore, the interest earned each year would be $990.
2. The recurrence relation to model the value of investment from year to year is:
Sn = Sn-1 + (r/100) * Sn-1
where Sn represents the value of the investment after n years, Sn-1 represents the value after n-1 years, and r represents the annual interest rate.
Using this recurrence relation, we can calculate the value of the investment for different years:
- S1 = 22,000 + 990 = 22,990
- S2 = 22,990 + (4.5/100) * 22,990 = 24,026.55
- S3 = 24,026.55 + (4.5/100) * 24,026.55 = 25,103.46
And so on...
3. To determine the value of the investment after 5 years, we can simply substitute n = 5 in the recurrence relation:
S5 = S4 + (r/100) * S4
= S3 + (r/100) * S3 + (r/100) * S3
= S2 + (r/100) * S2 + (r/100) * S2 + (r/100) * S2
= S1 + (r/100) * S1 + (r/100) * S1 + (r/100) * S1 + (r/100) * S1
Substituting values from previous calculations:
S1 = 22,000 + 990 = 22,990
So,
S5 = 22,990 + (4.5/100) * 22,990 + (4.5/100) * 22,990 + (4.5/100) * 22,990 + (4.5/100) * 22,990
= $27,037.44
Therefore, the value of the investment after 5 years would be $27,037.44.