Sequence Word Problems worksheet - Free Printable

Let's solve each problem step by step.

---

Problem 1:

Lauren opened a bank account with an initial $500. She is depositing $45.20 per week into her bank account. How many money will she have in her bank account after \( n \) weeks?

#### Solution:
Lauren starts with an initial amount of $500 and deposits $45.20 every week. The total amount in her bank account after \( n \) weeks can be expressed as:
\[
\text{Total Amount} = \text{Initial Amount} + (\text{Weekly Deposit} \times n)
\]
Substituting the given values:
\[
\text{Total Amount} = 500 + 45.20n
\]

#### Final Answer:
\[
\boxed{500 + 45.20n}
\]

---

Problem 2:

Zariah’s 100 meter dash times for her first four races were 14 seconds, 13.4 seconds, 12.8 seconds, and 12.2 seconds. Assuming race times will decrease at the same rate, what will the time for her 8th race be?

#### Solution:
The race times form an arithmetic sequence where:
- First term (\( a \)) = 14 seconds
- Common difference (\( d \)) = 13.4 - 14 = -0.6 seconds

The general formula for the \( n \)-th term of an arithmetic sequence is:
\[
a_n = a + (n-1)d
\]
We need to find the time for her 8th race (\( a_8 \)):
\[
a_8 = 14 + (8-1)(-0.6)
\]
\[
a_8 = 14 + 7(-0.6)
\]
\[
a_8 = 14 - 4.2
\]
\[
a_8 = 9.8
\]

#### Final Answer:
\[
\boxed{9.8}
\]

---

Problem 3:

Three yeast cells were placed in a laboratory dish at 2:00 p.m. The number of yeast cells doubles every ten minutes. Calculate the number of cells in the dish after one hour.

#### Solution:
The number of yeast cells doubles every 10 minutes. After one hour (60 minutes), the number of 10-minute intervals is:
\[
\frac{60}{10} = 6
\]
If the initial number of cells is 3, the number of cells after \( n \) intervals is given by:
\[
\text{Number of Cells} = \text{Initial Number} \times 2^n
\]
Here, \( n = 6 \):
\[
\text{Number of Cells} = 3 \times 2^6
\]
\[
2^6 = 64
\]
\[
\text{Number of Cells} = 3 \times 64 = 192
\]

#### Final Answer:
\[
\boxed{192}
\]

---

Problem 4:

At a temperature of 20°C, the common amoeba reproduces by splitting in half every 24 hours. If we start with a single amoeba, how many will there be after 8 days?

#### Solution:
The amoeba splits in half every 24 hours, meaning the number of amoebas doubles every day. After \( n \) days, the number of amoebas is given by:
\[
\text{Number of Amoebas} = \text{Initial Number} \times 2^n
\]
Here, the initial number is 1, and \( n = 8 \):
\[
\text{Number of Amoebas} = 1 \times 2^8
\]
\[
2^8 = 256
\]
\[
\text{Number of Amoebas} = 256
\]

#### Final Answer:
\[
\boxed{256}
\]

---

Problem 5:

Jack practices the gaiter every day. The first day he practices for 5 minutes. The next day he practices for 10 minutes. The following days he increases his practice by 5 minutes. How many days will it have taken him to practice for a total of 1 hour and 15 minutes from the first day?

#### Solution:
Jack's practice times form an arithmetic sequence where:
- First term (\( a \)) = 5 minutes
- Common difference (\( d \)) = 5 minutes

The total practice time after \( n \) days is the sum of the first \( n \) terms of this arithmetic sequence. The formula for the sum of the first \( n \) terms is:
\[
S_n = \frac{n}{2} \left( 2a + (n-1)d \right)
\]
We need the total practice time to be 1 hour and 15 minutes, which is:
\[
1 \text{ hour } 15 \text{ minutes} = 60 + 15 = 75 \text{ minutes}
\]
Substitute \( S_n = 75 \), \( a = 5 \), and \( d = 5 \) into the sum formula:
\[
75 = \frac{n}{2} \left( 2(5) + (n-1)(5) \right)
\]
\[
75 = \frac{n}{2} \left( 10 + 5(n-1) \right)
\]
\[
75 = \frac{n}{2} \left( 10 + 5n - 5 \right)
\]
\[
75 = \frac{n}{2} \left( 5n + 5 \right)
\]
\[
75 = \frac{n}{2} \cdot 5(n + 1)
\]
\[
75 = \frac{5n(n + 1)}{2}
\]
Multiply both sides by 2:
\[
150 = 5n(n + 1)
\]
Divide both sides by 5:
\[
30 = n(n + 1)
\]
This is a quadratic equation:
\[
n^2 + n - 30 = 0
\]
Factorize the quadratic equation:
\[
(n + 6)(n - 5) = 0
\]
The solutions are:
\[
n = -6 \quad \text{or} \quad n = 5
\]
Since \( n \) must be a positive integer, we have:
\[
n = 5
\]

#### Final Answer:
\[
\boxed{5}
\]

---

Final Answers:

1. \(\boxed{500 + 45.20n}\)
2. \(\boxed{9.8}\)
3. \(\boxed{192}\)
4. \(\boxed{256}\)
5. \(\boxed{5}\)