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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.