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Mathematics Grade IX
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Ch-6 Progressions
School Revise · Grade 9 · Advanced Maths (Optional) · Chapter 6
Exploring Some More Progressions
Mathematics at Advanced Level, Class 9. This optional chapter works further with the geometric progression, finds the sum of its first terms, and meets the surprising sum of a geometric series that goes on forever.
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What this chapter is about
In a geometric progression each term is the one before multiplied by a fixed number. In this chapter we recall that pattern, find a formula for the sum of the first n terms, and see how an unending progression can still add up to a finite total.
1. The geometric progression
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Geometric progression a geometric progression, GP, has each term equal to the one before it times a fixed common ratio r It runs a, ar, ar², ar³, and so on, where a is the first term. The ratio of any term to the one before is always r. In real life: Money at compound interest grows as a geometric progression, each year’s amount being the last multiplied by a fixed factor. |
2. The sum of a geometric progression
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Sum of the first n terms Sₙ = a(1 − rⁿ) / (1 − r), for r not equal to 1 Rather than adding every term, this formula gives the total at once from the first term a, the ratio r and the number of terms n. It is derived by subtracting r times the sum from the sum itself. In real life: It quickly totals a repayment plan or a savings scheme where each instalment grows by a fixed factor, without adding term by term. |
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Worked example 1 Question. Find the sum 2 + 4 + 8 + 16 + 32.
Answer: The sum is 62. |
3. A sum that never ends
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Sum to infinity if the common ratio satisfies |r| < 1, an unending GP sums to S = a / (1 − r) When each term is a fraction of the one before, the terms shrink so fast that even infinitely many of them add to a finite number. This works only when |r| is less than 1. In real life: A bouncing ball that rises to a fixed fraction of its last height travels a finite total distance, though it bounces endlessly in theory. |
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Worked example 2 Question. Find the sum of 1 + ½ + ¼ + ⅛ + … forever.
Answer: The sum is 2. |
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Worked example 3 Question. Find the sum of the first 4 terms of 3, 6, 12, 24.
Answer: The sum is 45. |
Practise with the interactive
Try this chapter hands on: change the values and watch the answer update live and animate. The interactive opens right here in the lesson.
Practice set A, multiple choice
1. In a GP, each term is the one before it …
Multiplied by a fixed common ratio r.
2. The sum of the first n terms of a GP is …
a(1 − rⁿ)/(1 − r), where a is the first term and r the common ratio.
3. An unending GP has a finite sum only when …
|r| < 1, so that the terms shrink quickly enough to add to a finite total.
4. For 5, 10, 20, 40, the common ratio r is …
2, since each term is twice the one before it.
Practice set B, short answer
1. Find the sum 3 + 9 + 27 + 81.
| 1 | a = 3, r = 3, n = 4. use Sₙ = a(1 − rⁿ)/(1 − r). |
| 2 | Substitute. 3(1 − 3⁴)/(1 − 3) = 3(1 − 81)/(−2). |
| 3 | Work it out. = 3 × (−80)/(−2) = 3 × 40 = 120. |
2. Find the sum of 1 + ⅓ + ⅛… where r = ⅓, forever.
| 1 | a = 1, r = ⅓, |r| < 1. use S = a/(1 − r). |
| 2 | Work it out. = 1/(1 − ⅓) = 1/(2/3) = 3/2. |
3. What is the common ratio of 16, 8, 4, 2?
½, because each term is half the one before it; since |r| < 1, an unending version would have a finite sum.
Quick summary
| Idea | The idea |
| GP | Each term times a fixed ratio r. |
| Terms | a, ar, ar², ar³, … |
| Sum of n terms | Sₙ = a(1 − rⁿ)/(1 − r). |
| Sum to infinity | a/(1 − r) when |r| < 1. |
| Compound interest | A real GP. |
| Common ratio | Any term divided by the one before. |
| Open the Virtual Lab |
These free Grade 9 Advanced Maths notes explain geometric progressions, summing a GP, and the infinite geometric series, with clear step by step worked examples and labelled diagrams for every student using the optional Advanced Level book.
© 2026 School Revise. All rights reserved. This lesson is original content written by School Revise, aligned to the CBSE Class 9 Mathematics at Advanced Level (Optional) syllabus. Unauthorised copying, reproduction or redistribution is not permitted. Curriculum names are used only to indicate alignment.
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