Geometric Sequence: Modeling Soda Consumption Reduction

Sequences & Series 9th-10th Grade
PROBLEM
Robert decided to gradually decrease the number of sodas he drank each month. He made this table to keep track: Month 1 has 150, Month 2 has 135, Month 3 has 121.5, and Month 4 has 109.35. Write a recursive formula for this situation and use your formula to determine how many sodas (to the nearest can) Robert can have in month 12.

Skills This Problem Builds

  • Pattern Recognition: Identifying geometric sequences from decimal data points
  • Recursive Thinking: Writing formulas that define each term using the previous term
  • Ratio Analysis: Computing and verifying common ratios in geometric sequences
  • Iterative Calculation: Applying recursive formulas repeatedly to reach distant terms
  • Real-World Modeling: Connecting mathematical sequences to exponential decay scenarios

Visualizing the Decay Pattern

Let's organize the given data and see the decreasing trend:

Month (n)Sodas f(n)Ratio f(n)/f(n-1)
1150—
2135135 ÷ 150 = 0.9
3121.5121.5 ÷ 135 = 0.9
4109.35109.35 ÷ 121.5 = 0.9
Robert decided to gradually decrease the number of sodas he drank each month. He made this table to keep track: Month...

The constant ratio of 0.9 confirms this is a geometric sequence where Robert reduces his consumption by 10% each month.

Solution: Method 1 — The Recursive Decay Formula

Step 1 — Identify the sequence type

Looking at the data, we need to check if this is arithmetic (constant difference) or geometric (constant ratio). Let's calculate the ratios between consecutive terms:

135 ÷ 150 = 0.9
121.5 ÷ 135 = 0.9
109.35 ÷ 121.5 = 0.9

Since the ratio is constant at 0.9, this is a geometric sequence where each month's consumption is 90% of the previous month.

Step 2 — Write the recursive formula

For a recursive formula, we need the first term and the rule for finding subsequent terms:

f(1) = 150
f(n) = 0.9 × f(n-1) for n ≥ 2

Step 3 — Apply the formula to find f(12)

We'll calculate each month step by step:

f(1) = 150
f(2) = 0.9 × 150 = 135
f(3) = 0.9 × 135 = 121.5
f(4) = 0.9 × 121.5 = 109.35
f(5) = 0.9 × 109.35 = 98.415
f(6) = 0.9 × 98.415 = 88.5735
f(7) = 0.9 × 88.5735 = 79.71615
f(8) = 0.9 × 79.71615 = 71.744535
f(9) = 0.9 × 71.744535 = 64.5700815
f(10) = 0.9 × 64.5700815 = 58.11307335
f(11) = 0.9 × 58.11307335 = 52.301766015
f(12) = 0.9 × 52.301766015 = 47.0715894135

Step 4 — Round to the nearest can

Since Robert can't drink a fraction of a can, we round 47.07 to the nearest whole number:

Robert can have 47 sodas in month 12.

Solution: Method 2 — The Explicit Formula Approach

Step 1 — Derive the explicit formula

For a geometric sequence with first term a and common ratio r, the explicit formula is:

f(n) = a × r^(n-1)

In our case, a = 150 and r = 0.9, so:

f(n) = 150 × (0.9)^(n-1)

Step 2 — Calculate f(12) directly

Instead of calculating each month sequentially, we can jump directly to month 12:

f(12) = 150 × (0.9)^(12-1)
f(12) = 150 × (0.9)^11
f(12) = 150 × 0.31384283767
f(12) = 47.076425651

Step 3 — Round to the nearest can

Rounding 47.076 to the nearest whole number gives us 47 sodas.

Why the explicit formula is faster: The recursive method requires calculating all intermediate values, while the explicit formula lets you compute any term directly. Both give the same answer, but explicit formulas are more efficient for finding distant terms.

Verification

Let's verify our answer using both approaches and check that our recursive formula produces the given data points:

Check given values:
f(1) = 150 ✓
f(2) = 0.9 × 150 = 135 ✓
f(3) = 0.9 × 135 = 121.5 ✓
f(4) = 0.9 × 121.5 = 109.35 ✓

Check f(12) with explicit formula:
f(12) = 150 × (0.9)^11 = 47.076... ≈ 47 ✓

We can also verify that 47 is reasonable: after 11 months of 10% decreases, Robert's consumption should be much less than his starting 150 sodas, but not close to zero yet. 47 sodas (about 31% of the original amount) fits this expectation perfectly.

Watch Out For These Mistakes

✗ Assuming arithmetic sequence: Looking at the differences: 150 - 135 = 15, 135 - 121.5 = 13.5. Since 15 ≠ 13.5, students might think there's no pattern or try to force an arithmetic formula like f(n) = f(n-1) - 15. This fails because the decrease is proportional, not constant.
✗ Using 0.1 instead of 0.9 as the ratio: Students might think "10% decrease means multiply by 0.1." Wrong! A 10% decrease means you keep 90%, so multiply by 0.9. Using 0.1 would give f(12) = 150 × (0.1)^11 ≈ 0.00000015 sodas, which is absurd.
✗ Forgetting to round at the end: The mathematical answer is 47.076 sodas, but Robert can't drink 0.076 of a soda. Always round to whole cans when the context involves discrete objects.
✗ Using the wrong base in the explicit formula: Writing f(n) = 150 × (0.9)^n instead of f(n) = 150 × (0.9)^(n-1). The first formula gives f(1) = 150 × 0.9 = 135, which doesn't match the given f(1) = 150. Always check your explicit formula against the first term.

The Pattern Behind This

This problem demonstrates exponential decay, one of the most important patterns in mathematics and science. The general structure is:

Recursive: f(1) = initial_value, f(n) = decay_factor × f(n-1)
Explicit: f(n) = initial_value × (decay_factor)^(n-1)

Key insights:

  • Constant percentage change always produces geometric sequences
  • Decay factor = 1 - percentage_lost (10% loss → factor of 0.9)
  • Half-life problems are special cases where the decay factor is 0.5
  • The math doesn't care about context — radioactive decay, population decline, and soda reduction follow identical formulas
Why the answer isn't clean: Exponential functions with non-integer bases rarely produce whole numbers. The decimal answer 47.076 is mathematically correct and normal — messiness is expected in real-world exponential models.

Real Applications

This exact mathematical structure appears throughout science and finance:

  • Radioactive decay: Carbon-14 decreases by 50% every 5,730 years, following f(t) = initial × (0.5)^(t/5730)
  • Medication clearance: Many drugs are eliminated from the body at a constant percentage per hour
  • Population decline: Endangered species populations often follow exponential decay models
  • Depreciation: Car values typically decrease by 10-20% per year in early years
  • Economic recession: GDP or employment can decline at constant percentage rates during downturns

What If?

1
Different Starting Point
Robert's friend Sarah decides to follow the same 10% monthly reduction pattern, but she starts with 200 sodas in month 1. Write the recursive formula for Sarah's consumption and find how many sodas she'll have in month 8.
Step 1 — Write Sarah's recursive formula

Same pattern, different starting value: f(1) = 200 and f(n) = 0.9 × f(n-1) for n ≥ 2.

Step 2 — Calculate each month

f(1) = 200
f(2) = 0.9 × 200 = 180
f(3) = 0.9 × 180 = 162
f(4) = 0.9 × 162 = 145.8
f(5) = 0.9 × 145.8 = 131.22
f(6) = 0.9 × 131.22 = 118.098
f(7) = 0.9 × 118.098 = 106.2882
f(8) = 0.9 × 106.2882 = 95.65938

Step 3 — Round to nearest can

Sarah will have 96 sodas in month 8.

Verification

Using explicit formula: f(8) = 200 × (0.9)^7 = 200 × 0.4782969 = 95.659 ✓

2
Adding a Constraint
Robert follows his original plan (starting at 150, reducing 10% monthly), but he decides he'll never go below 25 sodas per month. In which month will his calculated consumption first drop to 25 or fewer sodas?
Step 1 — Set up the inequality

We need 150 × (0.9)^(n-1) ≤ 25

Step 2 — Solve algebraically

(0.9)^(n-1) ≤ 25/150 = 1/6 ≈ 0.1667
Taking natural log: (n-1) × ln(0.9) ≤ ln(0.1667)
(n-1) × (-0.1054) ≤ -1.7918
n-1 ≥ 17.0 (inequality flips when dividing by negative)
n ≥ 18

Step 3 — Verify by calculation

f(17) = 150 × (0.9)^16 = 150 × 0.1853 = 27.80 (still above 25)
f(18) = 150 × (0.9)^17 = 150 × 0.1668 = 25.02 (still above 25)
f(19) = 150 × (0.9)^18 = 150 × 0.1501 = 22.52 (below 25)

Answer

Month 19 is when Robert's consumption first drops to 25 or fewer sodas.

3
Different Decay Rate
Michael wants to be more aggressive and reduce his soda consumption by 15% each month, starting with 180 sodas. Write both the recursive and explicit formulas for Michael's plan, and find his consumption in month 6.
Step 1 — Identify the decay factor

15% reduction means keeping 85% each month, so the decay factor is 0.85.

Step 2 — Write both formulas

Recursive:f(1) = 180 and f(n) = 0.85 × f(n-1) for n ≥ 2
Explicit:f(n) = 180 × (0.85)^(n-1)

Step 3 — Calculate month 6 using explicit formula

f(6) = 180 × (0.85)^5
f(6) = 180 × 0.4437
f(6) = 79.87

Step 4 — Round to nearest can

Michael will have 80 sodas in month 6.

Verification with recursive method

f(1) = 180, f(2) = 153, f(3) = 130.05, f(4) = 110.54, f(5) = 93.96, f(6) = 79.87 ✓

4
Reverse Engineering
Lisa follows the same type of reduction pattern (constant percentage decrease monthly). She started with some number of sodas in month 1, and by month 5 she was down to 64 sodas. If her monthly decay factor is 0.8 (20% reduction), how many sodas did she start with?
Step 1 — Set up the equation

Using the explicit formula: f(5) = initial × (0.8)^(5-1) = initial × (0.8)^4
We know f(5) = 64, so: 64 = initial × (0.8)^4

Step 2 — Calculate (0.8)^4

(0.8)^4 = 0.8 × 0.8 × 0.8 × 0.8 = 0.4096

Step 3 — Solve for initial value

64 = initial × 0.4096
initial = 64 ÷ 0.4096 = 156.25

Step 4 — Round and verify

Lisa started with 156 sodas.
Check: 156 × (0.8)^4 = 156 × 0.4096 = 63.9 ≈ 64 ✓

Frequently Asked Questions

How do you identify if a sequence is geometric or arithmetic? +
Check if consecutive terms have a constant ratio (geometric) or constant difference (arithmetic). In this problem, 135÷150 = 0.9 and 121.5÷135 = 0.9, showing a constant ratio of 0.9, so it's geometric with a 10% monthly decrease.
What's the difference between recursive and explicit formulas for sequences? +
A recursive formula defines each term using the previous term, like f(n) = 0.9 × f(n-1). An explicit formula gives any term directly from its position, like f(n) = 150 × (0.9)^(n-1). Both describe the same sequence but with different computational approaches.
Why do real-world exponential decay problems often give messy decimal answers? +
Exponential functions with non-integer bases rarely produce whole numbers. Here, 150 × (0.9)^11 = 36.98 sodas for month 12. Since you can't drink 0.98 of a soda, we round to 37 cans - this messiness is normal and expected in decay models.
DN

Dr. Neven Jurkovic

Mathematics Professor · Sequence & Series Specialist

NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-10-23

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