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Geometric Sequences and Series Worked Examples cheat sheet - grade 10-12

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Math Grade 10-12

Geometric Sequences and Series Worked Examples Cheat Sheet

A printable reference covering geometric sequence formulas, common ratio, explicit and recursive rules, finite and infinite series, and worked examples for grades 10-12.

TopicsAlgebraExponentsRatios and ProportionsPatternsFunctionsMathematicsStudy Guide
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Geometric sequences and series appear whenever a quantity changes by the same multiplicative factor each step. This cheat sheet helps students recognize geometric patterns, write formulas, and solve common worked-example problems quickly. It is especially useful for comparing explicit rules, recursive rules, finite sums, and infinite sums in one place.

Key Facts

  • A sequence is geometric if each term is found by multiplying the previous term by the same common ratio rr.
  • The common ratio is r=anan−1r = \frac{a_n}{a_{n-1}} for any consecutive nonzero terms.
  • The explicit formula for a geometric sequence is an=a1rn−1a_n = a_1 r^{n-1}, where a1a_1 is the first term.
  • The recursive formula for a geometric sequence is an=ran−1a_n = r a_{n-1} with a given starting value a1a_1.
  • The finite geometric series sum is Sn=a1(1−rn)1−rS_n = \frac{a_1(1-r^n)}{1-r} when r≠1r \ne 1.
  • An equivalent finite sum formula is Sn=a1(rn−1)r−1S_n = \frac{a_1(r^n-1)}{r-1} when r≠1r \ne 1.
  • An infinite geometric series converges only when ∣r∣<1|r| < 1, and its sum is S∞=a11−rS_{\infty} = \frac{a_1}{1-r}.
  • If r=1r = 1, every term is equal to a1a_1, so the finite sum is Sn=na1S_n = n a_1.

Vocabulary

Geometric sequence
A sequence in which each term is multiplied by the same common ratio to get the next term.
Common ratio
The constant multiplier rr between consecutive terms of a geometric sequence.
Explicit formula
A formula such as an=a1rn−1a_n = a_1 r^{n-1} that gives any term directly from its position number.
Recursive formula
A formula such as an=ran−1a_n = r a_{n-1} that defines each term using the previous term.
Finite geometric series
The sum of a limited number of terms in a geometric sequence.
Convergent infinite series
An infinite series with a finite sum, which occurs for geometric series when ∣r∣<1|r| < 1.

Common Mistakes to Avoid

  • Using addition instead of multiplication to find the pattern is wrong because geometric sequences have a constant ratio, not a constant difference.
  • Writing an=a1rna_n = a_1 r^n is wrong for the usual first-term formula because the exponent should be n−1n-1, so the correct form is an=a1rn−1a_n = a_1 r^{n-1}.
  • Finding rr by dividing the earlier term by the later term is wrong if the formula uses r=anan−1r = \frac{a_n}{a_{n-1}}; reversing the order gives the reciprocal ratio.
  • Using S∞=a11−rS_{\infty} = \frac{a_1}{1-r} when ∣r∣≥1|r| \ge 1 is wrong because the infinite geometric series does not converge in that case.
  • Forgetting parentheses in powers is wrong when the ratio is negative because (−2)4(-2)^4 and −24-2^4 have different meanings.

Practice Questions

  1. 1 A geometric sequence has a1=5a_1 = 5 and r=3r = 3. Find a6a_6 using an=a1rn−1a_n = a_1 r^{n-1}.
  2. 2 Find the sum of the first 88 terms of the geometric series with a1=2a_1 = 2 and r=4r = 4.
  3. 3 An infinite geometric series has a1=12a_1 = 12 and r=13r = \frac{1}{3}. Find S∞S_{\infty}.
  4. 4 Explain why the infinite geometric series with first term 77 and common ratio −1.2-1.2 does not have a finite sum.

Understanding Geometric Sequences and Series Worked Examples

A geometric sequence is best understood as repeated scaling. Adding the same amount creates a linear pattern, but multiplying by the same factor creates a pattern that can change very fast. A ratio greater than one produces growth.

A positive ratio between zero and one produces decay. A negative ratio makes the terms switch signs while their sizes grow or shrink. For example, starting at eighty and multiplying by one half gives eighty, forty, twenty, and ten.

Starting at eighty and multiplying by negative one half gives eighty, negative forty, twenty, negative ten. The sign pattern is important when checking an answer.

The term number matters because the first term is assigned number one, not number zero. This is why the exponent in an explicit rule is one less than the term number. For the first term, the exponent must be zero, and any nonzero number raised to the zero power is one.

This check catches many index mistakes. A recursive rule is useful when building a table one term at a time. An explicit rule is more useful for finding a distant term, such as term twenty, without calculating every earlier term.

Students should practice moving between a list, a recursive rule, and an explicit rule. Each form describes the same pattern from a different angle.

A series means that the terms are being added, so it answers a different kind of problem from a sequence. A sequence might describe the amount paid each month. A series describes the total paid after several months.

When finding a finite sum, first identify the first term, the ratio, and the number of included terms. Then check whether the answer should be positive or negative and whether its size makes sense. A ratio above one can make later terms much larger than the first term.

In that case, the total should usually be larger than the final term when all terms are positive. A ratio of one needs separate treatment because every term is identical.

Infinite sums can seem strange because infinitely many terms are involved. They only settle to one fixed total when each new term becomes smaller in size. A bouncing ball gives a useful model.

If each bounce reaches a fixed fraction of the previous height, the total distance can approach a limited value even though the ball keeps bouncing in the model. Negative ratios can converge too, because the partial totals move back and forth by smaller amounts. Do not assume that a sequence approaching zero guarantees every related calculation is simple.

Check the absolute size of the ratio first. If it is at least one, the terms do not shrink enough for an infinite geometric sum to exist. In worked examples, write the known values before choosing a formula and keep track of parentheses when powers or negative ratios appear.