MathematicsChapter 4: Sequences and Series
Sum of A Geometric Series
Finite GP: S_n = a (r^n - 1)/(r-1) if r != 1.
Learning objectives
- State the main ideas and notation used in sum of a geometric series.
- Apply the rules of sum of a geometric series in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Finite GP: Sn = a rn - 1r-1 if r != 1. If r=1, Sn = n a. If |r|<1 you can also write Sn = a 1-rn1-r (same thing). Meaning: factor a from a + a r + ... + a rn-1. The formula fails at r=1 because you would divide by 0; that case is just adding a, n times.
Mathematical form
a=3, r=2, n=5
S5 = 3 32-12-1
Sn = a rn - 1r-1 if r
Sn = n a. If
Where / symbols
- SUM OF A GEOMETRIC SERIES
- Finite GP: S_n = a (r^n - 1)/(r-1) if r != 1. If r=1, S_n = n a. If |r|<1 you can also write S_n = a (1-r^n)/(1-r) (same thing). Meaning: factor a from a + a r + ... + a r^{n-1}. The formula fails at r=1 because you would divide by 0; that case is just adding a, n times.
- 1. r NOT 1
- Write r^n carefully (signs if r negative). If r=-1, the sum flips a, 0, a, 0, ... depending on n even/odd.
- 2. r = 1
- Every term is a. Sum is n a. Do not plug r=1 into (r^n-1)/(r-1). Limit idea is for infinite GP next.
Relationship
- Sn = arn-1r-1, r!=1
- r=1: Sn = n a
Overview
Finite sum, then r=1:
r NOT 1
Write r^n carefully (signs if r negative). If r=-1, the sum flips a, 0, a, 0, ... depending on n even/odd.
Examples S_n formula · Negative r: alternate · Check with 3 terms by hand
r = 1
Every term is a. Sum is n a. Do not plug r=1 into rn-1r-1. Limit idea is for infinite GP next.
Examples 1+1+1 (n times) · Formula hole at r=1 · Separate case
WHAT — SUM OF A GEOMETRIC SERIES
Finite GP: Sn = a rn - 1r-1 if r != 1. If r=1, Sn = n a. If |r|<1 you can also write Sn = a 1-rn1-r (same thing). Meaning: factor a from a + a r + ... + a rn-1. The formula fails at r=1 because you would divide by 0; that case is just adding a, n times.
RULE — 1. r NOT 1
Write r^n carefully (signs if r negative). If r=-1, the sum flips a, 0, a, 0, ... depending on n even/odd.
METHOD — 2. r = 1
Every term is a. Sum is n a. Do not plug r=1 into rn-1r-1. Limit idea is for infinite GP next.
WORKED EXAMPLE
Worked: 3 + 6 + 12 + 24 + 48. a=3, r=2, n=5 S5 = 3 32-12-1 = 3*31 = 93
How to answer a sum of a geometric series question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- 3 + 6 + 12 + 24 + 48.
- a=3, r=2, n=5
- S5 = 3 32-12-1
- = 3*31 = 93
In simple words
If r=1 the main formula is illegal. Use n a.
Key points
- Sn = arn-1r-1, r!=1
- r=1: Sn = n a
- factor a
- just n copies
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Sum of A Geometric Series
Sticky Notes · Quick Revision
Core Idea
- Finite GP: Sn = a rn - 1r-1 if r != 1.
- If r=1, Sn = n a.
- If |r|<1 you can also write Sn = a 1-rn1-r (same thing).
1. r NOT 1
- Write r^n carefully (signs if r negative).
- If r=-1, the sum flips a, 0, a, 0, ...
- depending on n even/odd.
2. r = 1
- Every term is a.
- Sum is n a.
- Do not plug r=1 into rn-1r-1.
- Limit idea is for infinite GP next.
Worked Example
- Worked: 3 + 6 + 12 + 24 + 48.
- a=3, r=2, n=5
- S5 = 3 32-12-1
- = 3*31 = 93
Common Misconception
- If r=1 the main formula is illegal. Use n a.
FBISE Exam Check
- State the rule for sum of a geometric series before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Finite GP: Sn = a rn - 1r-1 if r != 1.
- If r=1, Sn = n a.
- If |r|<1 you can also write Sn = a 1-rn1-r (same thing).
- If r=1 the main formula is illegal. Use n a.
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