Geometric progression

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MODULE: Pre-Calculus (Functions and Series)

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✖️ 1. Defining geometric sequences via recursive definition and calculating the common ratio

🔁 Defining Geometric Sequences

  • A geometric sequence multiplies the previous term by a fixed number to get the next term.
  • The recursive rule is an=an−1⋅ra_n = a_{n-1} \cdot r where rr is the common ratio.
  • To find rr, divide any term by the term before it: r=anan−1r = \frac{a_n}{a_{n-1}}.
  • The common ratio stays the same throughout the entire sequence.
  • If a1=3a_1 = 3 and r=2r = 2, then a2=3⋅2=6a_2 = 3 \cdot 2 = 6, a3=6⋅2=12a_3 = 6 \cdot 2 = 12, and so on.

Example: Sequence 5, 15, 45, 135 has r=155=3r = \frac{15}{5} = 3.

💡 Each term = previous term × same multiplier

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1. Defining geometric sequences via recursive definition and calculating the common ratio

Geometric Sequences and the Common Ratio

A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a fixed nonzero constant called the common ratio rr. The recursive definition is an=an−1⋅ra_n = a_{n-1} \cdot r for n≥2n \geq 2, where a1a_1 is the first term.

Intuition: Each term is a scaled version of its predecessor, creating a multiplicative pattern rather than an additive one.

Core Rules:

  • The common ratio rr is calculated as r=anan−1r = \frac{a_n}{a_{n-1}} for any consecutive terms where an−1≠0a_{n-1} \neq 0.
  • rr must be constant throughout the sequence; if the ratio varies, the sequence is not geometric.
  • r≠0r \neq 0 (otherwise the sequence becomes trivial after the first term).
  • The first term a1a_1 and rr completely determine the entire sequence.

Consequence: Knowing any term and the common ratio allows reconstruction of all subsequent terms through repeated multiplication.

Example: In the sequence 3, 6, 12, 24, we compute r=63=2r = \frac{6}{3} = 2. Each term is twice the previous one.

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Find the common ratio rr for the geometric sequence: 7,28,112,4487, 28, 112, 448.

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✖️ 2. Interpreting sequence behavior based on the common ratio

📊 Behavior Based on Common Ratio

  • When ∣r∣>1|r| > 1, the sequence shows exponential growth (terms get larger in size).
  • When ∣r∣<1|r| < 1, the sequence shows exponential decay (terms shrink toward zero).
  • When rr is negative, the sequence alternates between positive and negative values.
  • When r=1r = 1, all terms stay constant (no growth or decay).
  • When r=−1r = -1, terms flip between two opposite values.

Example: For r=−0.5r = -0.5 starting at 8, the sequence is 8, -4, 2, -1, 0.5 (alternating decay).

💡 Negative rr = zigzag pattern, ∣r∣|r| controls size change

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2. Interpreting sequence behavior based on the common ratio

Behavior Classification by Common Ratio

The value of rr determines the qualitative behavior of a geometric sequence. When ∣r∣>1|r| > 1, terms grow exponentially in magnitude (growth). When ∣r∣<1|r| < 1, terms shrink toward zero (decay).

Intuition: The magnitude of rr controls whether terms expand or contract, while its sign controls whether terms alternate in sign.

Core Rules:

  • Exponential growth: ∣r∣>1|r| > 1 causes terms to increase in absolute value without bound.
  • Exponential decay: 0<∣r∣<10 < |r| < 1 causes terms to approach zero asymptotically.
  • Alternating oscillation: r<0r < 0 makes consecutive terms alternate in sign, creating a zigzag pattern.
  • Special cases: r=1r = 1 yields a constant sequence; r=−1r = -1 produces alternating values of equal magnitude.

Consequence: The sign and magnitude of rr together predict long-term sequence behavior, critical for modeling real phenomena.

Example: For a1=5a_1 = 5 and r=−0.5r = -0.5, the sequence is 5, -2.5, 1.25, -0.625, showing both decay and oscillation.

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A geometric sequence has a common ratio of r=0.75r = 0.75. Which of the following best describes the long-term behavior of its terms?

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✖️ 3. Using the explicit formula for the n-th term

🎯 Explicit Formula for Any Term

  • The explicit formula jumps directly to the nn-th term: an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}.
  • Here a1a_1 is the first term, rr is the common ratio, and nn is the position.
  • The exponent is n−1n-1 because the first term has zero multiplications applied.
  • This formula skips all intermediate calculations.

Example: Find the 6th term when a1=2a_1 = 2 and r=3r = 3. Then a6=2⋅36−1=2⋅243=486a_6 = 2 \cdot 3^{6-1} = 2 \cdot 243 = 486.

💡 Jump to any term = first term × ratio raised to (position - 1)

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3. Using the explicit formula for the n-th term

Explicit Formula for the n-th Term

The explicit formula an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} directly computes the nn-th term of a geometric sequence without calculating all preceding terms. Here a1a_1 is the first term, rr is the common ratio, and nn is the term position (with n≥1n \geq 1).

Intuition: The exponent n−1n-1 counts how many times we multiply a1a_1 by rr to reach the nn-th term.

Core Rules:

  • The formula applies for any positive integer nn.
  • The exponent is n−1n-1 because the first term requires zero multiplications by rr.
  • Domain: n∈Nn \in \mathbb{N} (natural numbers starting from 1).
  • This formula is derived by repeatedly applying the recursive definition.

Consequence: Direct computation eliminates the need for iterative calculation, enabling efficient access to distant terms.

Example: For a1=3a_1 = 3 and r=2r = 2, the 5th term is a5=3⋅25−1=3⋅16=48a_5 = 3 \cdot 2^{5-1} = 3 \cdot 16 = 48.

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A geometric sequence has a first term of a1=2a_1 = 2 and a common ratio of r=3r = 3. Calculate the 4th term of this sequence.

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✖️ 4. Calculating the sum of a finite geometric series

➕ Sum of Finite Geometric Series

  • To add the first nn terms, use the formula Sn=a11−rn1−rS_n = a_1 \frac{1 - r^n}{1 - r} when r≠1r \neq 1.
  • This formula works for any value of rr except exactly 1.
  • When r=1r = 1, the sum is simply Sn=n⋅a1S_n = n \cdot a_1 (just add the same number nn times).
  • The numerator 1−rn1 - r^n captures how the ratio compounds over nn terms.

Example: Sum first 4 terms of 3, 6, 12, 24. Here a1=3a_1 = 3, r=2r = 2, so S4=31−241−2=3−15−1=45S_4 = 3 \frac{1 - 2^4}{1 - 2} = 3 \frac{-15}{-1} = 45.

💡 Sum formula = first term × (ratio pattern over nn steps) ÷ (1 - ratio)

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4. Calculating the sum of a finite geometric series

Sum of a Finite Geometric Series

The sum of the first nn terms of a geometric sequence is given by Sn=a11−rn1−rS_n = a_1 \frac{1-r^n}{1-r} for r≠1r \neq 1. This formula aggregates all terms from a1a_1 to ana_n efficiently.

Intuition: The formula exploits the self-similar structure of geometric sequences to collapse the sum into a closed form.

Core Rules:

  • Restriction: The formula is valid only when r≠1r \neq 1; if r=1r = 1, then Sn=n⋅a1S_n = n \cdot a_1 (arithmetic sum).
  • An equivalent form is Sn=a1rn−1r−1S_n = a_1 \frac{r^n - 1}{r - 1}, useful when r>1r > 1 to avoid negative denominators.
  • The formula is derived by multiplying the sum by rr and subtracting to eliminate intermediate terms.
  • Domain: n∈Nn \in \mathbb{N}, r∈R∖[1]r \in \mathbb{R} \setminus [1].

Consequence: This enables rapid computation of cumulative totals in finance, physics, and computer science.

Example: For a1=2a_1 = 2, r=3r = 3, n=4n = 4: S4=21−341−3=2−80−2=80S_4 = 2 \frac{1-3^4}{1-3} = 2 \frac{-80}{-2} = 80.

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Find the sum of the first 4 terms of a geometric sequence where a1=5a_1 = 5 and r=2r = 2.

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✖️ 5. Applications in epidemiology and compound interest

🌍 Real-World Applications

  • Viral spread: If each infected person infects rr others, total cases grow geometrically each generation.
  • Compound interest: Money grows by factor (1+rate)(1 + \text{rate}) each period, forming a geometric sequence.
  • Use an=a1⋅rn−1a_n = a_1 \cdot r^{n-1} to predict future values after nn time steps.
  • Use SnS_n to calculate total accumulated quantity over nn periods.

Example: 1000 dollars at 5% annual interest for 3 years gives a3=1000⋅(1.05)3−1=1000⋅1.1025=1102.50a_3 = 1000 \cdot (1.05)^{3-1} = 1000 \cdot 1.1025 = 1102.50 dollars.

💡 Geometric sequences model anything that multiplies by a constant factor repeatedly

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5. Applications in epidemiology and compound interest

Modeling Viral Spread and Compound Interest

Geometric progressions model discrete exponential processes where quantities multiply by a fixed factor per time step. In epidemiology, if each infected person spreads disease to rr new people per generation, total infections follow an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}. In finance, compound interest applies the formula A=P(1+i)tA = P(1 + i)^t where PP is principal, ii is interest rate per period, and tt is the number of periods.

Intuition: Repeated multiplicative growth or decay naturally produces geometric sequences in real-world systems.

Core Rules:

  • Viral spread: rr represents the reproduction number; r>1r > 1 indicates epidemic growth, r<1r < 1 indicates decline.
  • Compound interest: Each period, the balance is multiplied by (1+i)(1 + i), making it a geometric sequence with ratio (1+i)(1 + i).
  • Both models assume discrete time steps and constant rates.

Consequence: These applications demonstrate how geometric progressions quantify exponential phenomena in biology and economics.

Example: 1000 dollars at 5% annual interest for 3 years yields 1000(1.05)3=1157.6251000(1.05)^3 = 1157.625 dollars.

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A public health team is monitoring a viral outbreak. They calculate that the reproduction number is r=0.85r = 0.85. Based on the geometric progression model for epidemiology, what does this indicate about the disease?

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