Geometric sequences & series
Practice Flashcards
What is a geometric sequence?
Track your progress — Sign up free to save your progress and get smart review reminders based on spaced repetition.
All Flashcards in Topic 1.3
Below are all 32 flashcards for this topic. Sign up free to track your progress and get personalized review schedules.
1.3.113 cards
What is a geometric sequence?
A sequence where each term is the previous one multiplied by a constant, the common ratio r. Example: 2, 6, 18, 54 has r = 3.
What is the common ratio, and how do you find it?
The constant multiplier: r = uₙ ÷ uₙ₋₁. Divide any term by the one before. Example: 12 ÷ 4 = 3.
A quantity 'increases by 8% each year' vs 'increases by 8 each year' — which model, and what is the key number?
'by 8%' multiplies ⇒ geometric, r = 1.08, uₙ = u₁rⁿ⁻¹. 'by 8' adds ⇒ arithmetic, d = 8. Words like percent / ratio / times / doubles signal geometric; a fixed amount signals arithmetic.
Why is it rⁿ⁻¹ and not rⁿ?
You start at u₁ and multiply by r only on each step after the first — (n − 1) times. Example: u₅ = u₁ r⁴.
How do you find r from two terms, e.g. u₂ = 6 and u₅ = 48?
Divide the values, then take the (steps)-th root: 48 ÷ 6 = 8 over 3 steps, so r = ∛8 = 2.
When can the common ratio be negative?
When the number of steps between the two terms is even, r = ±(root of the value-ratio). An odd number of steps gives a unique r.
When are three terms u₁, u₂, u₃ geometric?
When the ratios are equal: u₂/u₁ = u₃/u₂, i.e. u₂² = u₁u₃ (middle squared = product of neighbours).
Find k if 4, k, 25 are geometric (k > 0).
k² = 4 × 25 = 100, so k = 10.
How is geometric different from arithmetic?
Arithmetic ADDS the same d each step; geometric MULTIPLIES by the same r. 3, 6, 9 is arithmetic; 3, 6, 12 is geometric.
Find u₆ for u₁ = 5 and r = 2.
u₆ = 5 × 2⁵ = 160.
Why does the middle term squared equal the product of its neighbours?
Because consecutive ratios are equal: u₂/u₁ = u₃/u₂. Cross-multiplying gives u₂² = u₁u₃.
Three expressions are geometric and the condition gives a quadratic. How many values of the unknown?
Up to two — solve the quadratic and report both. Use any stated condition (e.g. all terms positive) to choose between them.
How do you avoid mixing up n and n − 1 in a geometric question?
Count the ×r jumps from the start — that count is the power. The nth TERM is n − 1 jumps (term 1 = 0 jumps); 'after n bounces / years' is n jumps (the start is counted). E.g. dropped 6 m, after the 4th bounce = 6(½)⁴ = 0.375.
1.3.211 cards
How do you spot that a question needs the geometric SUM (Sₙ), not the nth term (uₙ)?
Look for a TOTAL — 'sum of', 'altogether', 'total saved' over terms that multiply by a constant ⇒ Sₙ = u₁(rⁿ − 1)/(r − 1). A single value ('the 8th term', 'value after 8 years') ⇒ uₙ = u₁rⁿ⁻¹.
Which form of the geometric sum should you use?
Either works. Use (rⁿ − 1)/(r − 1) when r > 1 and (1 − rⁿ)/(1 − r) when 0 < r < 1 to keep the numbers positive.
What do you need to use the geometric sum formula?
u₁, r and n. If r isn't given, find it first from two terms.
Find S₅ for 3 + 6 + 12 + … .
u₁ = 3, r = 2: S₅ = 3(2⁵ − 1)/(2 − 1) = 3 × 31 = 93.
How do you find the smallest n with Sₙ past a target?
Set Sₙ > target and solve for n; on Paper 2 scan the GDC table of Sₙ and round up to the next whole number.
Why does the geometric sum formula need r ≠ 1?
If r = 1 every term equals u₁, so the sum is just n × u₁ (and the formula would divide by zero).
Find S₄ for u₁ = 6, r = ½.
S₄ = 6(1 − 0.5⁴)/(1 − 0.5) = 6(0.9375)/0.5 = 11.25.
Sₙ = 2(3ⁿ − 1). Find u₁ and r.
Compare with u₁(rⁿ − 1)/(r − 1): r = 3 and u₁/(3 − 1) = 2 ⇒ u₁ = 4.
On Paper 2, how do you sum a geometric series on the GDC?
Use sum(seq(u₁ r^(x−1), x, 1, n)) or read a table of Sₙ. Round n up for 'smallest n' questions.
How do you show a geometric sum equals a given closed form like a(bⁿ − 1)?
Substitute u₁ and r into Sₙ = u₁(rⁿ − 1)/(r − 1) and simplify until it matches. E.g. u₁ = 4, r = 3 → 4(3ⁿ − 1)/2 = 2(3ⁿ − 1).
How do you find the total distance a dropped ball travels over n bounces?
Distance = the first drop + 2 × (sum of the rebound heights). The drop counts once; every rebound is travelled up and down. Use the finite geometric sum Sₙ for the rebound heights.
1.3.38 cards
How do you model compound interest as a geometric sequence?
Each period the balance multiplies by r = 1 + (rate as a decimal). After n periods: balance = start × rⁿ.
What is the common ratio for x% growth? For x% decay?
Growth: r = 1 + x/100. Decay: r = 1 − x/100. E.g. 6% growth → 1.06; 15% decay → 0.85.
How do you find how long until an amount doubles?
Solve rⁿ = 2 (logs) or use the GDC/TVM solver; round n up to the next whole period.
$2000 at 6% per year — when does it first exceed $4000?
2000 × 1.06ⁿ > 4000 → 1.06ⁿ > 2 → n ≈ 11.9 → 12 years.
On the TI-84 TVM solver, how do you find the years to a target?
Enter I% = rate, PV = −start, PMT = 0, FV = target, P/Y = C/Y = periods per year, then solve for N. Money out is negative.
How is depreciation different from growth?
Depreciation is decay: r = 1 − rate (0 < r < 1), so the value shrinks by a fixed percentage each period.
Why is compound interest not the same as simple interest?
Compound multiplies the growing balance by r each period (geometric); simple adds a fixed amount each period (arithmetic).
A machine worth $20 000 loses 15%/yr. Value after 4 years?
r = 0.85; 20 000 × 0.85⁴ ≈ $10 440.
Topic 1.3 study notes
Full notes & explanations for Geometric sequences & series
Math AA SL exam skills
Paper structures, command terms & tips
Want smart review reminders?
Sign up free to track your progress. Our spaced repetition algorithm will tell you exactly which cards to review and when.
Start Free