Eleventh Grade · Math · week 31

Geometric sequences and series

Find terms and sums of geometric sequences and series, including infinite geometric series.

A geometric sequence multiplies by the same fixed ratio, r, to get from one term to the next -- the multiplying cousin of last week's adding sequence. Its explicit formula is a_n = a_1 * r^(n-1), and its finite sum formula is S_n = a_1(1-r^n)/(1-r), for r not equal to 1.

Something remarkable happens when |r| < 1 and you add up INFINITELY many terms: each term shrinks toward zero, and the partial sums creep up on a fixed limiting value rather than growing forever. That limit has its own tidy formula: S = a_1/(1-r). If |r| >= 1, the terms do not shrink, the sum grows without bound, and no finite total exists.

📋 For the teacher
Teach it (10–15 min):
  • Contrast this week's "multiply" move against last week's "add" move explicitly, side by side, using the exact same starting number for both -- the difference in growth speed is the whole point.
  • Introduce the infinite sum with a physical demonstration if possible (repeatedly halving a strip of paper) so "infinitely many terms, finite total" feels concrete rather than paradoxical.
  • For the bouncing-ball problem, draw the trip as a timeline (down, up-down, up-down, ...) so it is clear why every bounce height (except the initial drop) gets counted TWICE.
  • Always check |r| < 1 explicitly before using the infinite sum formula -- applying it when |r| >= 1 gives a nonsense answer for a sum that does not actually exist.
Talk about it:
  • How is a geometric sequence different from an arithmetic sequence in how consecutive terms relate?
  • Why does an infinite geometric series need |r| < 1 to have a finite sum?
  • In the bouncing ball problem, why is each bounce height counted twice but the initial drop only once?

Mastery looks like: They find terms and finite sums of a geometric sequence, and correctly apply the infinite sum formula only when |r| < 1.

Common stumbles: Using the infinite sum formula when |r| >= 1; forgetting to double the bounce heights (up and down) in a real-world total-distance problem.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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