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Key ideas at a glance

A geometric sequence has a constant ratio r between consecutive terms, and r is not zero.
If r is positive, all terms keep one sign. If r is negative, the signs alternate.
Geometric means are the terms inserted between two given numbers so that the whole list is geometric.
The sum to infinity exists only when −1 < r < 1.

Geometric_Sequence_and_Infinite_Series→Key ideas at a glance

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Rules to remember

Comparison aspect	Main idea	Keywords
n-th term	Tₙ = a × rⁿ⁻¹
Sum of n terms	Sₙ = a(1 − rⁿ) ÷ (1 − r) when r ≠ 1
Sum to infinity	S∞ = a ÷ (1 − r) when \|r\| < 1
Geometric means	if k means are inserted between A and B the ratio is r = (B ÷ A)^{1/(k + 1)}	insertedbetweenratio
Three terms in G.P.	if a c are in G.P then b² = ac	then

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Convergence table

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One short example for each rule

Comparison aspect	Main idea	Keywords
•	0.777… = 0.7 + 0.07 + … with a = 0.7 and r = 0.1. So S∞ = 0.7 ÷ 0.9 = 7/9
Rule 6 (bouncing ball).	A ball dropped from 16 m rebounds to half its previous height. Find the total distance	droppedreboundshalfprevious
First drop	16 m. Rebounds: 8 … m each followed by a fall of the same height. Total = 16 + 2 × [8 ÷ (1 − 1/2)] = 16 + 32 = 48 m	Reboundsfollowedfallsame
Rule 7 (divergent case).	Explain why 1 + 2 + 4 + 8 + … has no sum to infinity	Explaininfinity
•	Here r = 2 so \|r\| ≥ 1. The terms grow without limit so the partial sums grow without limit too	Heretermsgrowwithout

Comparison aspect	Main idea	Keywords
Rule 1 (compound growth).	1000 is invested at 5% a year compounded yearly. Find the value after 3 years	1000investedyearcompounded
•	The values form a G.P. with a = 1000 and r = 1.05. T₄ = 1000 × 1.05³ = 1157.63	valuesform10001157
Rule 2 (decay).	A sample of 80 g halves every hour. How much remains after 3 hours?	samplehalveseveryhour
•	a = 80 and r = 1/2. T₄ = 80 × (1/2)³ = 10 g
Rule 3 (geometric means).	Insert two geometric means between 3 and 24	Insertbetween
Four terms	24. Then r³ = 24 ÷ 3 = 8 so r = 2. The means are 6 and 12	Thenmeans
Rule 4 (sum to infinity).	Find the sum of 16 + 8 + 4 + 2 + …	Find
•	a = 16 and r = 1/2. S∞ = 16 ÷ (1 − 1/2) = 32
Rule 5 (repeating decimal).	Write 0.777… as a fraction	Writefraction

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Lesson summary

Find r by dividing any term by the term before it. Check that the same r works for the next pair.
Growth and decay problems (interest, depreciation, half-life) are geometric with the factor per period as r.
Use S∞ = a/(1 − r) only when |r| < 1. Otherwise, state that the infinite sum does not exist.
In a bouncing problem, the first drop is added once and each bounce is counted up and down.

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Lesson 2 Summary: Geometric Sequences, Series and Applications