Sum of a Geometric Series

Sum of a Geometric Series

What is it?

Finite GP: Sn = a rⁿ - 1r-1 if r != 1. If r=1, Sn = n a. If |r|<1 you can also write S_n = a 1-rⁿ1-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.

In simple words

• Finite GP: S_n = a rⁿ - 1r-1 if r != 1.
• If r=1, S_n = n a.
• If |r|<1 you can also write S_n = a 1-rⁿ1-r (same thing).
• If r=1 the main formula is illegal. Use n a.

Important terms

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 S_n = a 1-rⁿ1-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 rⁿ-1r-1. Limit idea is for infinite GP next.

What you should learn

• 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.

Examples

• S_n formula
• Negative r: alternate
• Check with 3 terms by hand

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.

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.

Full school-level explanation

Finite GP: Sn = a rⁿ - 1r-1 if r != 1. If r=1, Sn = n a. If |r|<1 you can also write S_n = a 1-rⁿ1-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.

Finite sum, then r=1:

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 S_n = a 1-rⁿ1-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.

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

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.

Remember

• factor a

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.

Remember

• just n copies

Examples

• 1+1+1 (n times)
• Formula hole at r=1
• Separate case

How to answer a sum of a geometric series question

How it works

• 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.

Remember

• S_n = arⁿ-1r-1, r!=1
• r=1: S_n = n a
• 93
• r=1 separate
• factor a
• just n copies
• S_n = arⁿ-1r-1, r!=1
• r=1: S_n = n a

How the ideas connect

Worked: 3 + 6 + 12 + 24 + 48. a=3, r=2, n=5 S₅ = 3 32-12-1 = 3*31 = 93

Deeper understanding

Go beyond memorising Sum of a Geometric Series. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: 3 + 6 + 12 + 24 + 48. a=3, r=2, n=5 S₅ = 3 32-12-1 = 3*31 = 93

Common mistakes

Exam focus

Core Idea

1. r NOT 1

2. r = 1

Worked Example

Exam focus

Quick revision

• S_n = arⁿ-1r-1, r!=1
• r=1: S_n = n a
• 93
• r=1 separate
• 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 S_n = a
• 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 ev

If r=1 the main formula is illegal. Use n a.

For long answers on Sum of a Geometric Series: define clearly → explain the main parts → give one example → state one exam caution.