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:
10000(1+0.035/4)^n
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) It's a geometric sequence, since the common ratio is 1+.035/4
Now just use what you know about GPs
To solve this problem, we need to use the given formula for the balance: 10000(1+0.035/4)^n. Let's find the answers to each question step by step.
a) To find the first five terms of the sequence, we substitute the values of n from 1 to 5 into the formula and calculate the corresponding balance for each quarter.
For n = 1:
Balance = 10000(1+0.035/4)^1 = 10000(1.00875) ≈ $10087.50
For n = 2:
Balance = 10000(1+0.035/4)^2 = 10000(1.01753) ≈ $10175.30
For n = 3:
Balance = 10000(1+0.035/4)^3 = 10000(1.02645) ≈ $10264.50
For n = 4:
Balance = 10000(1+0.035/4)^4 = 10000(1.03551) ≈ $10355.10
For n = 5:
Balance = 10000(1+0.035/4)^5 = 10000(1.04472) ≈ $10447.20
Therefore, the first five terms of the sequence are approximately: $10087.50, $10175.30, $10264.50, $10355.10, and $10447.20.
b) To identify the type of sequence, we observe the formula. In this case, the exponent (n) is continuously increasing, meaning it is an exponential sequence. Additionally, the sequence is increasing as more quarters pass, so it is also a growing or increasing exponential sequence.
c) To find the balance in the account after 30 quarters, we substitute n = 30 into the formula:
Balance = 10000(1+0.035/4)^30 = 10000(1.33100) ≈ $13310.00
Therefore, the balance in the account after 30 quarters will be approximately $13,310.00.
a) To find the first 5 terms, we can substitute different values of n into the formula:
For n = 1:
Balance = 10000(1 + 0.035/4)^1 = 10102.50
For n = 2:
Balance = 10000(1 + 0.035/4)^2 = 10207.63
For n = 3:
Balance = 10000(1 + 0.035/4)^3 = 10315.95
For n = 4:
Balance = 10000(1 + 0.035/4)^4 = 10427.65
For n = 5:
Balance = 10000(1 + 0.035/4)^5 = 10542.91
The first 5 terms of the sequence are: 10102.50, 10207.63, 10315.95, 10427.65, 10542.91.
b) This is an exponential sequence. The formula for the balance in the account after each quarter involves an exponent, which indicates exponential growth or decay.
c) To find the balance after 30 quarters, we substitute n = 30 into the formula:
Balance = 10000(1 + 0.035/4)^30
Calculating this value, the balance in the account after 30 quarters is approximately $16,304.95.