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Infinite Gp and Recurring Decimals
If |r|<1, the infinite sum S = a1-r. If |r|>=1 the infinite GP does not settle to a finite school sum (except the special a=0). A recurring decimal is a GP in disguise: 0.333... = 310 + 3100 + ... with r=110. Meaning: each extra term is a smaller bite, so the tail vanishes when |r|<1.
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Infinite Gp and Recurring Decimals
What is it?
If |r|<1, the infinite sum S = a1-r. If |r|>=1 the infinite GP does not settle to a finite school sum (except the special a=0). A recurring decimal is a GP in disguise: 0.333... = 310 + 3100 + ... with r=110. Meaning: each extra term is a smaller bite, so the tail vanishes when |r|<1.
In simple words
- If |r|<1, the infinite sum S = a1-r.
- If |r|>=1 the infinite GP does not settle to a finite school sum (except the special a=0).
- A recurring decimal is a GP in disguise: 0.333...
- Infinite formula needs |r|<1. 1+2+4+... is not 11-2.
Important terms
INFINITE GP AND RECURRING DECIMALS: If |r|<1, the infinite sum S = a1-r. If |r|>=1 the infinite GP does not settle to a finite school sum (except the special a=0). A recurring decimal is a GP in disguise: 0.333... = 310 + 3100 + ... with r=110. Meaning: each extra term is a smaller bite, so the tail vanishes when |r|<1.
1. INFINITE SUM: Need |r|<1. 1+12+14+... = 2. If |r|>1 terms blow up. If r=-12, still |r|<1, S = a1-r.
2. 0.abc repeating: 0.3 = 3/101-110 = 13. Split non-repeating prefix if 0.12\dot{3} (school method: two infinites or 10^k tricks). Show the first repeating block as a.
What you should learn
- State the main ideas and notation used in infinite gp and recurring decimals.
- Apply the rules of infinite gp and recurring decimals in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- a1-r
- First term a
- Not for |r|>=1
INFINITE SUM
Need |r|<1. 1+12+14+... = 2. If |r|>1 terms blow up. If r=-12, still |r|<1, S = a1-r.
0.abc repeating
0.3 = 3/101-110 = 13. Split non-repeating prefix if 0.12\dot{3} (school method: two infinites or 10^k tricks). Show the first repeating block as a.
Recommended for Class 11
Full school-level explanation
If |r|<1, the infinite sum S = a1-r. If |r|>=1 the infinite GP does not settle to a finite school sum (except the special a=0). A recurring decimal is a GP in disguise: 0.333... = 310 + 3100 + ... with r=110. Meaning: each extra term is a smaller bite, so the tail vanishes when |r|<1.
When it converges, then decimals:
WHAT — INFINITE GP AND RECURRING DECIMALS
If |r|<1, the infinite sum S = a1-r. If |r|>=1 the infinite GP does not settle to a finite school sum (except the special a=0). A recurring decimal is a GP in disguise: 0.333... = 310 + 3100 + ... with r=110. Meaning: each extra term is a smaller bite, so the tail vanishes when |r|<1.
RULE — 1. INFINITE SUM
Need |r|<1. 1+12+14+... = 2. If |r|>1 terms blow up. If r=-12, still |r|<1, S = a1-r.
METHOD — 2. 0.abc repeating
0.3 = 3/101-110 = 13. Split non-repeating prefix if 0.12\dot{3} (school method: two infinites or 10^k tricks). Show the first repeating block as a.
WORKED EXAMPLE
Worked: 0.222... a = 210, r = 110 S = 2101-110 = 15910 = 29 check: 29 = 0.222...
INFINITE SUM
Need |r|<1. 1+12+14+... = 2. If |r|>1 terms blow up. If r=-12, still |r|<1, S = a1-r.
Remember
- |r|<1 only
Examples
- a1-r
- First term a
- Not for |r|>=1
0.abc repeating
0.3 = 3/101-110 = 13. Split non-repeating prefix if 0.12\dot{3} (school method: two infinites or 10^k tricks). Show the first repeating block as a.
Remember
- geometric tail
Examples
- 0.555...=59
- 0.2727...=2799
- Prefix then GP
How to answer a infinite gp and recurring decimals 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
- |r|<1: S=a1-r
- repeating decimal -> infinite GP
- 29
- |r|<1
- |r|<1 only
- geometric tail
- |r|<1: S=a1-r
- repeating decimal -> infinite GP
How the ideas connect
Worked: 0.222... a = 210, r = 110 S = 2101-110 = 15910 = 29 check: 29 = 0.222...
Deeper understanding
Go beyond memorising Infinite Gp and Recurring Decimals. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: 0.222... a = 210, r = 110 S = 2101-110 = 15910 = 29 check: 29 = 0.222...
Common mistakes
Exam focus
Core Idea
1. INFINITE SUM
2. 0.abc repeating
Worked Example
Exam focus
Quick revision
- |r|<1: S=a1-r
- repeating decimal -> infinite GP
- 29
- |r|<1
- INFINITE GP AND RECURRING DECIMALS: If |r|<1, the infinite sum S = a1-r. If |r|>=1 the infinite GP does not settle to a finite scho
- 1. INFINITE SUM: Need |r|<1. 1+12+14+... = 2. If |r|>1 terms blow up. If r=-12, still |r|<1, S = a1-r.
Infinite formula needs |r|<1. 1+2+4+... is not 11-2.
For long answers on Infinite Gp and Recurring Decimals: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- |r|<1: S=a1-r
- repeating decimal -> infinite GP
- 29
- |r|<1
Important formulas
- |r|<1: S=a1-r
- repeating decimal -> infinite GP
- 29
Common exam traps
- Infinite formula needs |r|<1. 1+2+4+... is not 11-2.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Infinite formula needs |r|<1. 1+2+4+... is not 11-2.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
Infinite formula needs |r|<1. 1+2+4+... is not 11-2.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
|r|<1: S=a1-r
repeating decimal -> infinite GP
29
|r|<1
If |r|<1, the infinite sum S = a1-r. If |r|>=1 the infinite GP does not settle to a finite school sum (except the special a=0). A recurring decimal is a GP
|r|<1: S=a1-r
- Definition
- If |r|<1, the infinite sum S = a1-r. If |r|>=1 the infinite GP does not settle to a finite school sum (except the special a=0). A recurring decimal is a GP
- Formula
- |r|<1: S=a1-r
- Example
- a1-r
- Critical warning
- Infinite formula needs |r|<1. 1+2+4+... is not 11-2.
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Exam Practice
Define
Q1. Define Infinite Gp and Recurring Decimals.
Explain
Q2. Explain: INFINITE SUM.
Example
Q3. Give one clear example of Infinite Gp and Recurring Decimals and say why it fits.
Mistake
Q4. State one common mistake students make about Infinite Gp and Recurring Decimals.
Exam note
Q5. Write a short exam note on Infinite Gp and Recurring Decimals: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.