Suppose you deposit $10,000 into an account earning 3.5% interest
compounded quarterly. After n quarters the balance in the account is given
by the formula:
a) Each quarter can be viewed as a term of a sequence. List the first 5 terms.
b) Identify the type of sequence this is. Explain.
c) Find the balance in the account after 30 quarters.
b = 10000 [1 + (.035/4)]^n
a) calculate
b) geometric
... the ratio (r) of any two consecutive terms is a constant
... r = 1 + .035/4
c) calculate
To solve this problem, we will use the formula for compound interest:
A = P(1 + r/n)^(n*t)
where:
A = final amount/balance
P = initial deposit/principal
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case:
P = $10,000
r = 3.5% = 0.035 (convert to decimal form)
n = 4 (compounded quarterly)
t = number of quarters
Now let's go through each part of the question step by step.
a) To find the first 5 terms, we'll calculate the balance after each quarter using the formula mentioned above.
First quarter:
A = 10,000(1 + 0.035/4)^(4*1)
A = 10,000(1.00875)^4
A ≈ 10,357.07
Second quarter:
A = 10,357.07(1 + 0.035/4)^(4*1)
A = 10,357.07(1.00875)^4
A ≈ 10,723.51
Third quarter:
A = 10,723.51(1 + 0.035/4)^(4*1)
A = 10,723.51(1.00875)^4
A ≈ 11,100.48
Fourth quarter:
A = 11,100.48(1 + 0.035/4)^(4*1)
A = 11,100.48(1.00875)^4
A ≈ 11,488.15
Fifth quarter:
A = 11,488.15(1 + 0.035/4)^(4*1)
A = 11,488.15(1.00875)^4
A ≈ 11,886.88
The first 5 terms of the sequence are approximately:
$10,357.07, $10,723.51, $11,100.48, $11,488.15, $11,886.88
b) This sequence is an example of exponential growth, specifically exponential growth with compound interest. As each term is calculated by multiplying the previous term by a constant factor (1 + r/n), the sequence shows exponential growth.
c) To find the balance in the account after 30 quarters, we'll use the formula mentioned earlier:
A = P(1 + r/n)^(n*t)
A = 10,000(1 + 0.035/4)^(4*30)
A ≈ 10,000(1.00875)^(120)
A ≈ $21,992.19
Therefore, the balance in the account after 30 quarters would be approximately $21,992.19.
To answer these questions, we need to understand how compound interest works and use the given formula to calculate the balance after a specific number of quarters. Let's go step by step:
a) To find the first 5 terms of the sequence, we need to substitute the values of n = 1, 2, 3, 4, 5 into the formula. The formula for the balance after n quarters is:
Balance = Principal * (1 + (interest rate / number of compounding periods))^n
In this case, the principal is $10,000, the interest rate is 3.5% (or 0.035 as a decimal), and the number of compounding periods is 4 (since it is compounded quarterly). Substituting these values into the formula, we can calculate the first 5 terms:
Term 1: Balance = $10,000 * (1 + (0.035 / 4))^1
Term 2: Balance = $10,000 * (1 + (0.035 / 4))^2
Term 3: Balance = $10,000 * (1 + (0.035 / 4))^3
Term 4: Balance = $10,000 * (1 + (0.035 / 4))^4
Term 5: Balance = $10,000 * (1 + (0.035 / 4))^5
Solving these equations will give us the first 5 terms of the sequence.
b) To identify the type of sequence, we need to observe the pattern in the terms. A sequence is classified based on its common difference or common ratio between the terms. In this case, since we are dealing with compound interest, the common ratio between the terms will be the factor (1 + (interest rate / number of compounding periods)).
If the value we calculated for the common ratio is the same for every term, then it is a geometric sequence. If the common difference between terms is constant, then it is an arithmetic sequence.
c) To find the balance in the account after 30 quarters, we can use the same formula mentioned earlier and substitute n = 30:
Balance = $10,000 * (1 + (0.035 / 4))^30
Solving this equation will give us the balance in the account after 30 quarters.