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noemir42
17.10.2020 •
Mathematics
On 1st January 2020, Laurie invests P dollars in an account that pays a nominal annual interest rate of 5.5%, compounded quarterly. The amount of money in Laurie’s account at the end of each year follows a geometric sequence with a common ratio, r. Find the value of r. Also, Laurie makes no further deposits to or withdrawals from the account. Find the year in which the amount of money in Laurie’s account will become double the amount she invested.
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Ответ:
Using compound interest and a geometric sequence, it is found that:
The common ratio isCompound interest:
![A(t) = P\left(1 + \frac{r}{n}\right)^{nt}](/tpl/images/0816/0008/03426.png)
A(t) is the amount of money after t years. P is the principal(the initial sum of money). r is the interest rate(as a decimal value). n is the number of times that interest is compounded per year. t is the time in years for which the money is invested or borrowed.In this problem:
Rate of 5.5%, henceHence, considering the end of each year, that is,
, the common ratio will be:
The nth term of a geometric sequence is given by:
In which A(0) is the initial value.
In this problem,
, and the time to double is t for which
, hence:
2020 + 12.69 = 20.32.69.
Hence, the amount of money will double in the year 2032.
A similar problem is given at link
Ответ:
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.
Ответ:
The numbers are 27 and 12
Step-by-step explanation:
Let the smaller number be x and the larger number be y.
If the difference between them is 15, then
Also , four times the larger number is nine times the smaller.
From equation (1),
We put the 3rd equation into the 2nd to get:
Expand to get:
Group similar terms
Therefore the larger number is