Geometric Series Patterns that multiply, formulas that amplify.

Key Idea

Geometric Sequence

A geometric sequence multiplies each term by a constant ratio \(r\). First term: \(a_1\). General term: \(a_n = a_1 r^{\,n-1}\).

Identify \(a_1\) and find \(r = \frac{a_2}{a_1}\) to describe any geometric sequence.

Find the Ratio r

Goal: practise extracting the common ratio from raw sequence data.

1

Spot Consecutive Terms

From the sequence \(3, 6, 12, 24, \dots\) pick pairs: \(3,6\); \(6,12\); \(12,24\).

2

Divide to Form Fractions

Compute \( \frac{6}{3}, \frac{12}{6}, \frac{24}{12} \).

3

Simplify ⇒ Common Ratio

Each fraction simplifies to 2, giving the common ratio \( r = 2 \).

Pro Tip:

If every consecutive pair gives the same quotient, the sequence is geometric.

General Term

\[a_n = a_1\, r^{(n-1)}\]

The nth-term formula lets you jump straight to any term in the sequence.

State it clearly and interpret each variable correctly.

Variable Definitions

\(a_1\) first term
\(r\) common ratio
\(n\) term position

Applications

Calculate distant terms

Find the 50th or 100th term quickly.

Model exponential growth

Describe interest, population or radioactive decay.

Multiple Choice Question

Question

For the geometric sequence with first term \(a_1 = 5\) and common ratio \(r = 3\), what is \(a_5\)?

1
135
2
405
3
243
4
625

Hint:

Use \(a_5 = 5 \times 3^{4}\).

Growth vs Decay

https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/UrXq1PCVcq3iRaOTg6nXfH9FAZg3PCj9KgiwgRZR.png

r = 2 (blue) grows, r = 0.5 (red) decays.

Compare r = 2 and r = 0.5

Both sequences begin at 1 on the graph.

With r = 2, each term doubles; the curve shoots upward, showing exponential growth.

With r = 0.5, each term halves; the curve sinks toward the x-axis, showing exponential decay.

Key Points:

  • Graphical intuition: steep upward slope means growth; downward slope means decay.
  • Comparison: identical starts diverge quickly because r differs.
  • Long-term: r > 1 → ∞, 0 < r < 1 → 0.

Partial Sum \(S_n\)

A finite geometric series sums the first \(n\) terms of a sequence. Know this formula by heart when \(r \neq 1\).

\[S_n = a_1\, \frac{1 - r^{n}}{1 - r}\quad\text{for}\; r \ne 1\]

Variable Definitions

\(S_n\) sum of first \(n\) terms
\(a_1\) first term
\(r\) common ratio
\(n\) number of terms

Applications

Loan schedules

Determine the remaining balance after \(n\) equal payments.

Population totals

Estimate cumulative population after \(n\) years of geometric growth.

Where Sn Comes From

1
\[S_n = a_1 + a_1 r + a_1 r^2 + \dots + a_1 r^{n-1}\]

Write the series in expanded form.

2
\[r S_n = a_1 r + a_1 r^2 + \dots + a_1 r^{n}\]

Multiply every term by \(r\).

3
\[S_n - r S_n = a_1 - a_1 r^{n}\]

Subtract to eliminate the middle terms.

4
\[S_n (1 - r) = a_1 (1 - r^{n})\]

Factor \(S_n\) on the left.

5
\[S_n = a_1 \frac{1 - r^{n}}{1 - r}\]

Divide by \(1 - r\) to isolate \(S_n\).

Key Insight:

Telescoping quickly collapses the series.

Key Takeaways

🔑

Multiply, don’t add

Each step scales by \(r\); think multiplication, not addition.

📍

an formula

Jump to term \(a_n = a_1 r^{n-1}\) instantly.

🧮

Sum Sn

Finite sum: \(S_n = a_1 \frac{1 - r^{n}}{1 - r}\).

♾️

Convergence rule

Infinite series converges only when \(|r| < 1\).

🌍

Real-world power

Governs compound interest, population change and exponential decay.