Geometric Sequences: Savings Account Growth Formula

What This Problem Teaches

• Geometric sequence recognition — Identifying when a sequence follows multiplicative rather than additive growth
• Explicit formula construction — Converting from recursive patterns to closed-form expressions
• Financial literacy foundations — Understanding compound growth and how small monthly rates accumulate over time
• Mathematical modeling — Translating real-world scenarios into algebraic expressions
• Exponential notation mastery — Working with bases and exponents in practical contexts

Solution: Method 1 — The Geometric Sequence Approach

This problem is asking us to find an explicit formula for a geometric sequence. Let's work through this step-by-step by identifying the pattern and applying the standard formula.

Step 1 — Identify the initial term and growth factor

Bailee starts with $500 as her initial deposit. Each month, her balance is multiplied by 1.0025. This means:

• Initial amount (when n = 0): a₀ = 500
• Growth factor: r = 1.0025

Step 2 — Trace the first few terms to confirm the pattern

Let's see what happens month by month:

• After 0 months: a₀ = 500
• After 1 month: a₁ = 500 × 1.0025 = 501.25
• After 2 months: a₂ = 501.25 × 1.0025 = 502.50
• After 3 months: a₃ = 502.50 × 1.0025 = 503.76

This confirms we have a geometric sequence where each term equals the previous term multiplied by 1.0025.

Step 3 — Apply the geometric sequence explicit formula

For any geometric sequence, the explicit formula is:

where a₀ is the initial term and r is the common ratio.

Step 4 — Substitute our specific values

Substituting a₀ = 500 and r = 1.0025:

This is our explicit formula describing Bailee's account balance after n months.

Solution: Method 2 — Percentage Growth Model

We can also approach this by recognizing that a multiplication factor of 1.0025 represents percentage growth. This perspective helps connect the mathematics to financial concepts.

Step 1 — Convert the factor to a growth rate

A factor of 1.0025 means:

• 1 represents keeping 100% of the original balance
• 0.0025 represents gaining an additional 0.25% each month

So Bailee's account grows by 0.25% monthly.

Step 2 — Apply compound growth formula

For compound growth, we use the formula:

Step 3 — Substitute the values

With initial amount = $500, growth rate = 0.0025, and time period = n months:

This matches our result from Method 1, confirming our formula is correct.

Verification

Let's verify our formula by checking it against our manually calculated values from earlier.

For n = 1:

a₁ = 500 × (1.0025)¹ = 500 × 1.0025 = 501.25 ✓

For n = 2:

a₂ = 500 × (1.0025)² = 500 × 1.00500625 = 502.503125 ≈ 502.50 ✓

For n = 0:

a₀ = 500 × (1.0025)⁰ = 500 × 1 = 500 ✓

Our formula correctly produces the expected values, confirming that aₙ = 500 × (1.0025)ⁿ is the correct explicit formula.

Common Pitfalls

The Underlying Pattern

This problem illustrates the general structure of geometric sequences in financial contexts. The explicit formula for any geometric sequence is:

Where:

• a₀ = initial value (principal amount)
• r = common ratio (growth factor per period)
• n = number of periods elapsed

In financial applications, r > 1 represents growth (like savings account interest), while 0 < r < 1 represents decay (like depreciation). The key insight is that when a quantity changes by a constant percentage each period, we get exponential growth or decay, not linear change.

How to Spot This Problem Type

Geometric sequence problems often contain these key phrases:

• "increases by a factor of" or "multiplied by" — signals multiplication, not addition
• "growth rate of" or "percentage increase" — when combined with compounding
• "each month/year/period" — indicates regular, repeating process
• "explicit formula" — asks for closed-form expression, not recursive formula

Watch out for problems that mention "compound interest," "population growth," "radioactive decay," or "appreciation/depreciation." These contexts almost always involve geometric sequences.

The key test: if each term equals the previous term times a constant, you have a geometric sequence. If each term equals the previous term plus a constant, you have an arithmetic sequence.

Real Applications

• Investment accounts: Mutual funds and retirement accounts often grow by a relatively consistent monthly factor, making this formula essential for financial planning.
• Population modeling: When birth rates are stable, populations grow geometrically. Epidemiologists use similar models to track disease spread.
• Technology scaling: Moore's Law (computer processing power doubling every two years) follows geometric growth, as do many technology adoption rates.