Answer:
1) The common ratio = 1.055
2) The year in which the amount of money in Laurie's account will become double is the year 2032
Step-by-step explanation:
1) The given information are;
The date Laurie made the investment = 1st, January, 2020
The annual interest rate of the investment = 5.5%
Type of interest rate = Compound interest
Therefore, we have;
The value, amount, of the investment after a given number of year, given as follows;
Amount in her account = a, a × (1 + i), a × (1 + i)², a × (1 + i)³, a × (1 + i)ⁿ
Which is in the form of the sum of a geometric progression, Sₙ given as follows;
Sₙ = a + a × r + a × r² + a × r³ + ... + a × rⁿ
Where;
n = The number of years
Therefore, the common ratio = 1 + i = r = 1 + 0.055 = 1.055
The common ratio = 1.055
2) When the money doubles, we have;
2·a = a × rⁿ = a × 1.055ⁿ
2·a = a × 1.055ⁿ
2·a/a = 2 = 1.055ⁿ
2 = 1.055ⁿ
Taking log of both sides gives;
㏒2 = ㏒(1.055ⁿ) = n × ㏒(1.055)
㏒2 = n × ㏒(1.055)
n = ㏒2/(㏒(1.055)) ≈ 12.95
The number of years it will take for the amount of money in Laurie's account to double = n = 12.95 years
Therefore, the year in which the amount of money in Laurie's account will become double = 2020 + 12..95 = 2032.95 which is the year 2032
The year in which the amount of money in Laurie's account will become double = year 2032.
