Arithmetic & Geometric Sequence Word Problems 4Q | PDF | Sequence ... - Free Printable

Let’s solve each problem step by step.

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Problem 1: Walking Program

The patient walks:
- Week 1: 5 km
- Week 2: 8 km
- Week 3: 11 km
- ... and so on for 10 weeks.

We need to find how far they walk in week 10, and answer all parts.

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

Look at the differences between terms:

8 - 5 = 3
11 - 8 = 3

The difference is constant (always +3). That means it’s an arithmetic sequence.

In a geometric sequence, you multiply by the same number each time — but here we’re adding, not multiplying.

✔ Answer: Arithmetic, because each term increases by a constant amount (3 km).

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b. Write out the sequence using blanks where appropriate.

Start with 5, then add 3 each time:

Week 1: 5
Week 2: 8
Week 3: 11
Week 4: ___ → 11 + 3 = 14
Week 5: ___ → 14 + 3 = 17
Week 6: ___ → 17 + 3 = 20
Week 7: ___ → 20 + 3 = 23
Week 8: ___ → 23 + 3 = 26
Week 9: ___ → 26 + 3 = 29
Week 10: ___ → 29 + 3 = 32

So the full 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:

> aₙ = a₁ + (n - 1)d

Where:
- aₙ = value of the nth term
- a₁ = first term = 5
- d = common difference = 3
- n = term number (week number)

Plug in:

> aₙ = 5 + (n - 1) × 3

You can simplify if you want:

aₙ = 5 + 3n - 3
→ aₙ = 3n + 2

Either form is okay, but let’s use the simplified one:
✔ aₙ = 3n + 2

Check: For n=1 → 3(1)+2 = 5 ✔️
n=2 → 3(2)+2 = 8 ✔️
Good!

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

Use the formula: aₙ = 3n + 2

Plug in n = 10:

a₁₀ = 3(10) + 2 = 30 + 2 = 32

Or using the other version:
a₁₀ = 5 + (10 - 1)×3 = 5 + 9×3 = 5 + 27 = 32

Same answer.

✔ So, 32 km.

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e. Write your final answer as a sentence.

During the 10th week, the patient will walk 32 kilometers.

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Problem 2: Virus Reproduction

A virus divides into two every growth period. Starts with 1 virus. After 10 divisions, how many viruses?

This means:
- Start: 1 virus
- After 1st division: 2 viruses
- After 2nd division: 4 viruses
- After 3rd: 8 viruses
- etc.

Each time, it doubles → multiplied by 2.

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

Look at the pattern:

1, 2, 4, 8, 16...

Each term is multiplied by 2 to get the next.

That’s a geometric sequence.

Arithmetic would be adding the same number — but here we’re multiplying.

✔ Answer: Geometric, because each term is found by multiplying the previous term by 2.

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

After 0 divisions: 1 virus (starting point)
After 1st division: 2
After 2nd: 4
After 3rd: 8
After 4th: ___ → 16
After 5th: ___ → 32
After 6th: ___ → 64
After 7th: ___ → 128
After 8th: ___ → 256
After 9th: ___ → 512
After 10th: ___ → 1024

First three terms after start: 2, 4, 8

But note: sometimes people count “after 0 divisions” as term 1. Let’s clarify.

If we say:
- Term 1 = after 0 divisions = 1
- Term 2 = after 1 division = 2
- Term 3 = after 2 divisions = 4
- ...
- Term 11 = after 10 divisions = ?

Wait — the question says: “starting with a single virus AFTER 10 divisions”

So we start at 1 (before any division), then after 1st division → 2, ..., after 10th division → ?

So the sequence of virus counts *after each division* is:

After 0 div: 1
After 1 div: 2
After 2 div: 4
...
After 10 div: ?

So the sequence has 11 terms if we include start, but the question asks for “after 10 divisions”, meaning we go through 10 doubling steps.

Let’s define:

Let a₀ = 1 (initial)
a₁ = 2 (after 1st division)
a₂ = 4
...
a₁₀ = ? (after 10th division)

So the sequence from a₀ to a₁₀ is geometric with ratio 2.

But for part b, it says “fill in the first three terms”. Probably meaning the first three values after starting.

So:

Term 1 (after 0 div): 1
Term 2 (after 1 div): 2
Term 3 (after 2 div): 4

Then blanks for rest up to after 10 divisions.

But to avoid confusion, let’s write the sequence of virus counts after each division step, including start:

Position: 0 → 1
Position: 1 → 2
Position: 2 → 4
Position: 3 → 8
Position: 4 → 16
Position: 5 → 32
Position: 6 → 64
Position: 7 → 128
Position: 8 → 256
Position: 9 → 512
Position: 10 → 1024

So for part b, writing the sequence with blanks:

1, 2, 4, ___, ___, ___, ___, ___, ___, ___, ___

Fill first three: already filled as 1, 2, 4.

Actually, the instruction says: “Fill in the first three terms.” So maybe they mean the first three non-starting? But better to include start.

Looking back: “Write out the sequence using blanks where appropriate. Fill in the first three terms.”

Probably they expect:

After division 0: 1
After division 1: 2
After division 2: 4
After division 3: ___
...
After division 10: ___

So sequence: 1, 2, 4, ___, ___, ___, ___, ___, ___, ___, ___

First three terms: 1, 2, 4 — already filled.

✔ So we leave it as: 1, 2, 4, _, _, _, _, _, _, _, _

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c. Write an explicit formula for the sequence.

Geometric sequence formula:

> aₙ = a₁ × r^(n-1)

But here, if we let n = number of divisions, then:

At n=0 divisions: a₀ = 1
At n=1: a₁ = 2
At n=2: a₂ = 4

So actually, it’s easier to use:

> aₙ = a₀ × rⁿ

Where:
- a₀ = initial amount = 1
- r = common ratio = 2
- n = number of divisions

So:

> aₙ = 1 × 2ⁿ = 2ⁿ

Check:
n=0 → 2⁰ = 1 ✔️
n=1 → 2¹ = 2 ✔️
n=2 → 4 ✔️
Perfect.

✔ Explicit formula: aₙ = 2ⁿ, where n is 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 formula: aₙ = 2ⁿ

n = 10

a₁₀ = 2¹⁰ = 1024

Calculate:
2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128
2⁸ = 256
2⁹ = 512
2¹⁰ = 1024

✔ So, 1024 viruses.

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e. Write your final answer as a sentence.

After 10 divisions, there will be 1024 viruses in the system.

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Final Answer:

Problem 1:
a. Arithmetic, because each term increases by a constant amount (3 km).
b. Sequence: 5, 8, 11, 14, 17, 20, 23, 26, 29, 32
c. Explicit formula: aₙ = 3n + 2
d. 32 km
e. During the 10th week, the patient will walk 32 kilometers.

Problem 2:
a. Geometric, because each term is found by multiplying the previous term by 2.
b. Sequence: 1, 2, 4, ___, ___, ___, ___, ___, ___, ___, ___ (first three terms filled)
c. Explicit formula: aₙ = 2ⁿ
d. 1024 viruses
e. After 10 divisions, there will be 1024 viruses in the system.