Sequences and Series word problems - Jason Murphy | Library ... - Free Printable

Let's solve each problem step-by-step using the required formulas and explanations.

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Problem 1: Walking Program

Given:
- Week 1: 5 km
- Week 2: 8 km
- Week 3: 11 km
- Pattern continues for 10 weeks

#### a. Is the sequence arithmetic or geometric? Explain your answer.

We check the difference between consecutive terms:
- $ 8 - 5 = 3 $
- $ 11 - 8 = 3 $

The common difference is constant (3), so this is an arithmetic sequence.

✔ Answer: The sequence is arithmetic because each term increases by a constant amount (3 km) from the previous term.

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#### b. Write out the sequence using blanks where appropriate.

We start with:
- Week 1: 5
- Week 2: 8
- Week 3: 11
- Week 4: 14
- Week 5: 17
- Week 6: 20
- Week 7: 23
- Week 8: 26
- Week 9: 29
- Week 10: 32

So, the sequence is:

$$
5,\ 8,\ 11,\ 14,\ 17,\ 20,\ 23,\ 26,\ 29,\ 32
$$

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#### c. Write an explicit formula for the sequence.

For an arithmetic sequence, the explicit formula is:

$$
a_n = a_1 + (n - 1)d
$$

Where:
- $ a_1 = 5 $ (first term)
- $ d = 3 $ (common difference)

So:

$$
a_n = 5 + (n - 1)(3)
$$

Simplify:

$$
a_n = 5 + 3n - 3 = 3n + 2
$$

✔ Explicit formula: $ a_n = 3n + 2 $

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#### d. How far will the patient walk during the 10th week? Show all work.

Use the formula:

$$
a_{10} = 3(10) + 2 = 30 + 2 = 32
$$

✔ Answer: The patient will walk 32 km in the 10th week.

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#### e. Write your final answer as a sentence.

✔ Answer: The patient will walk 32 kilometers during the 10th week.

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Problem 2: Virus Reproduction

Given:
- A single virus divides into two every time.
- After each division, the number doubles.
- We want to know how many viruses there are after 10 divisions.

Starts with 1 virus.

#### a. Is the sequence arithmetic or geometric? Explain your answer.

Let’s look at the pattern:
- Division 0 (start): 1
- Division 1: 2
- Division 2: 4
- Division 3: 8

Each term is multiplied by 2 → common ratio of 2

This is a geometric sequence.

✔ Answer: The sequence is geometric because each term is multiplied by a constant factor (2) to get the next term.

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#### b. Write out the sequence using blanks where appropriate. Fill in the first three terms.

Sequence:
- Division 0: 1
- Division 1: 2
- Division 2: 4
- Division 3: 8
- ...

So:

$$
1,\ 2,\ 4,\ \_,\ \_,\ \_,\ \_,\ \_,\ \_,\ \_
$$

But since we're asked for the first three terms, we write:

$$
1,\ 2,\ 4,\ \dots
$$

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#### c. Write an explicit formula for the sequence.

For a geometric sequence, the explicit formula is:

$$
a_n = a_1 \cdot r^{n-1}
$$

Where:
- $ a_1 = 1 $ (starting with one virus)
- $ r = 2 $ (each division doubles the count)

So:

$$
a_n = 1 \cdot 2^{n-1} = 2^{n-1}
$$

✔ Explicit formula: $ a_n = 2^{n-1} $

> Note: Here, $ n $ represents the number of divisions.

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#### d. How many viruses will be in a system starting with a single virus AFTER 10 divisions?

Use the formula:

$$
a_{10} = 2^{10 - 1} = 2^9 = 512
$$

✔ Answer: There will be 512 viruses after 10 divisions.

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#### e. Write your final answer as a sentence.

✔ Answer: After 10 divisions, there will be 512 viruses in the system.

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✔ Final Answers Summary:

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Problem 1:

- a. Arithmetic — constant difference of 3 km.
- b. Sequence: $ 5,\ 8,\ 11,\ 14,\ 17,\ 20,\ 23,\ 26,\ 29,\ 32 $
- c. Explicit formula: $ a_n = 3n + 2 $
- d. $ a_{10} = 3(10) + 2 = 32 $
- e. The patient will walk 32 kilometers during the 10th week.

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Problem 2:

- a. Geometric — each term is multiplied by 2.
- b. Sequence: $ 1,\ 2,\ 4,\ \dots $
- c. Explicit formula: $ a_n = 2^{n-1} $
- d. $ a_{10} = 2^{9} = 512 $
- e. After 10 divisions, there will be 512 viruses in the system.

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