Geometric Sequence (n-th Term)

Statement

A geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a fixed constant called the common ratio. If the first term is \(a\) and the common ratio is \(r\), then the \(n\)-th term is

\[ u_n = ar^{\,n-1}. \]

Why it works

• The first few terms are \(u_1=a,\; u_2=ar,\; u_3=ar^2,\; u_4=ar^3,\ldots\)
• We can see the pattern: to reach the \(n\)-th term we have multiplied by \(r\) exactly \(n-1\) times, so \(u_n=ar^{n-1}\).
• When \(|r|>1\) the terms grow in magnitude (exponential growth). When \(|r|<1\) they shrink towards 0 (exponential decay). If \(r<0\) the terms alternate signs.

Recipe (Steps to Apply)

1. Identify the first term \(a\) (usually the first number listed).
2. Find the common ratio \(r\) by dividing a term by the one before it: \(r=\dfrac{u_{k+1}}{u_k}\) (for any consecutive pair).
3. Substitute into \(u_n = ar^{n-1}\) to find any term.
4. To find a particular position \(n\) for a known value, set \(u_n\) equal to that value and solve \(ar^{n-1}= \text{value}\) (often by recognising powers or using logs if needed).

Spotting it

Look for a constant multiplicative change between consecutive terms (e.g., ×2 each time, or ×0.5 each time). That is geometric, not arithmetic (which uses constant addition).

Common pairings

• Sum of first \(n\) terms of a geometric series.
• Exponential growth/decay models (interest, population, depreciation).
• Indices and powers rules.

Mini examples

1. Sequence: \(3, 6, 12, 24, \ldots\) → \(a=3, r=2\). Then \(u_5=3\cdot 2^{4}=48\).
2. Sequence: \(80, 40, 20, 10, \ldots\) → \(a=80, r=\tfrac12\). Then \(u_n=80\left(\tfrac12\right)^{n-1}\).

Pitfalls

• Mixing up geometric with arithmetic sequences. Check by dividing, not subtracting.
• Forgetting the exponent is \(n-1\), not \(n\).
• Sign errors when \(r\) is negative (terms alternate signs).

Exam Strategy

• Always write \(a\) and \(r\) clearly before substituting into \(u_n=ar^{n-1}\).
• For “find \(n\)” questions, try to recognise powers first (e.g., \(2^k, 3^k, 10^k\)).
• Check reasonableness: with \(0<r<1\) terms should decrease; with \(r>1\) they should increase.

Worked Micro Example

Given the geometric sequence \(5, 10, 20, 40, \ldots\). Find the 7th term.

Here \(a=5,\; r=2\). So \(u_7=5\cdot 2^{6}=5\cdot 64=320.\)

Summary

The n-th term of a geometric sequence with first term \(a\) and common ratio \(r\) is \(u_n=ar^{n-1}\). Use it to compute terms, identify the formula from data, and compare growth/decay behaviours.