Find geometric progression terms and sums
Use the nth-term and finite-sum formulas to solve geometric progression problems with positive or negative ratios.
Section 12 · Series
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the nth-term formula for a GP?
What you will learn
- Find the nth term of a geometric progression
- Find the sum of its first n terms
- Apply the formulas when the common ratio is negative or fractional
Each term is multiplied by the same ratio
For first term a and common ratio r, uₙ = arⁿ⁻¹. The exponent is n − 1 because the first term uses zero multiplications by r.
A negative ratio alternates signs, and a fractional ratio can make the terms shrink. Keep parentheses around a negative r when using powers.
A finite geometric sum has a ratio-dependent formula
For r ≠ 1, Sₙ = a(rⁿ − 1)/(r − 1), equivalently a(1 − rⁿ)/(1 − r). The formula is supplied in the 0606 formula list.
When r = 1, every term is a and Sₙ = na. For contextual questions, identify the first term and common ratio before applying the formula.
Pause and explain
What is the common ratio of 2, −6, 18, −54?
Worked example
Find the fifth term and the sum of the first five terms of 3, 6, 12, 24, …
Show the worked solution
- Here a = 3, r = 2 and n = 5.
- u₅ = 3 × 2⁴ = 48.
- S₅ = 3(2⁵ − 1)/(2 − 1) = 3 × 31 = 93.
Answer Fifth term 48; sum 93.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For the GP 2, −6, 18, … find the sixth term and sum the first six terms.
Give me a hint
Use r = −3, keeping the sign inside powers.
Compare my reasoning
- u₆ = 2(−3)⁵ = −486.
- S₆ = 2(1 − (−3)⁶)/(1 − (−3)).
- This is 2(1 − 729)/4 = −364.
Sixth term −486; sum −364.
Look for these in your work
- Negative ratio retained
- Term exponent is n − 1
- Finite sum sign checked
For the GP 81, 27, 9, … find the fifth term and the sum of the first five terms.
Give me a hint
The common ratio is 1/3.
Compare my reasoning
- u₅ = 81(1/3)⁴ = 1.
- S₅ = 81[1 − (1/3)⁵]/[1 − 1/3].
- The five terms sum to 81 + 27 + 9 + 3 + 1 = 121.
Fifth term 1; sum 121.
Look for these in your work
- Fractional ratio applied
- Term checked from the sequence
- Sum checked independently
A culture starts with 20 cells and doubles each week. How many cells are present in week 8, and how many week-counted observations from week 1 through week 8 total?
Give me a hint
Use a = 20, r = 2 and interpret week 1 as the first term.
Compare my reasoning
- Week 8 is u₈ = 20 × 2⁷ = 2560.
- The total of weeks 1 through 8 is S₈ = 20(2⁸ − 1)/(2 − 1).
- Evaluate 20 × 255.
Week 8: 2560 cells; total across weeks 1–8: 5100 cell-counts.
Look for these in your work
- Week indexing starts at the first term
- Finite total matches the stated weeks
- Context units kept clear
Common mistakes
- Using n rather than n − 1 in the term exponent
- Confusing the ratio with the common difference
- Dropping brackets around a negative common ratio
- Using the finite-sum formula when r = 1 without the special case
What can you explain now?
What is the fourth term of a GP with a = 5 and r = 2?
Compare with the explanation
40
u₄ = 5 × 2³ = 40.
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Make it stick
How do the nth-term and finite-sum formulas for a GP depend on a and r?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.