Sequences and series
Finite geometric series worksheets
Sum geometric lists, evaluate finite sigma notation and find the number of terms from a first term, last term and total. Choose integers, tenths or mixed practice.
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Infinite geometric series worksheets
Decide convergence or divergence, find infinite geometric sums and determine common ratios from terms, sigma notation or first-term and total givens. Choose fractions or decimals.
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Determine whether each series converges, evaluate its sum when one exists, and find the common ratio. Lists continue with the same ratio.
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Decide when an infinite sum exists
Each sheet has four convergence questions, four infinite-sum questions and four common-ratio questions. Choose fractions and whole numbers or decimals and whole numbers.
Read a continuing list, a sigma expression or the stated first term and ratio. Some ratio questions give the first term and a finite total instead. Decimal terms stay exact; repeating sums retain an exact fraction with any approximation clearly marked.
Learning objective
What this activity builds
Connect the size of a common ratio with convergence, an exact infinite sum and inverse ratio calculations.
Worked example
Find the sum of 3 + 3/2 + 3/4 + 3/8 + ... .
6
The common ratio is 1/2, whose absolute value is less than 1. The infinite sum is 3 / (1 - 1/2) = 6.
Teaching notes
Make the practice count
- For a nonzero first term, convergence requires the absolute value of the ratio to be less than 1.
- Ratios of 1 or -1 do not produce a finite infinite sum.
- Use S = a₁ / (1 - r) only after checking convergence.
- Given a first term and a finite total, rearrange the sum formula to r = 1 - a₁ / S.
- Decimal prompts retain exact terminating terms; any approximation in a key is explicitly marked.
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Choose fractions or decimals. Decide convergence, find an infinite sum when one exists and determine the common ratio. Exact fractions are retained when a decimal sum repeats.
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Sum finite arithmetic lists, evaluate sigma notation and find the number of terms from a first term, last term and total. Choose integers or tenths.
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