Geometric sequences
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What makes a sequence geometric?
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All Flashcards in Topic 1.3
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1.3.18 cards
What makes a sequence geometric?
A sequence is geometric if you multiply by the same number each step. That fixed multiplier is the common ratio r.
Think: same multiplier
How do you find the common ratio r?
Divide any term by the term before it: r = uₙ₊₁ ÷ uₙ.
Divide, not subtract
What is the nth-term formula for a geometric sequence?
uₙ = u₁ · rⁿ⁻¹
Starts from u₁
If r = 0.5, does the sequence grow or shrink?
It shrinks. When 0 < r < 1, each term is a fraction of the one before it.
0<r<1
If a geometric sequence has negative r, what pattern do the signs follow?
The signs alternate. For example, r = −2 gives 4, −8, 16, −32, ...
Signs flip
Sequence 3, 6, 12, 24, ... What are u₁ and r?
u₁ = 3 and r = 2, because each term is multiplied by 2.
Read first term + multiplier
What does n mean in a geometric-sequence question?
n is the position number of the term. uₙ is the value of that term.
Position vs value
If 384 = 3 · 2ⁿ⁻¹ and 128 = 2⁷, what should you do next?
Match the exponents: n − 1 = 7, so n = 8.
Same base -> same exponent
1.3.28 cards
What is the difference between a geometric sequence and a geometric series?
A sequence is the list of terms. A series is what you get when you add those terms together.
List vs sum
What is the formula for the sum of the first n terms of a geometric series?
Sₙ = a(1 − rⁿ) ÷ (1 − r), for r ≠ 1.
Finite geometric sum
In Sₙ = a(1 − rⁿ)/(1 − r), what does a mean?
a is the first term of the geometric sequence.
First term
When should you use a sum formula instead of the nth-term formula?
Use the sum formula when the question wants the total of several terms, not just one term.
Total or one term?
What common mistake happens if a student uses uₙ when the question wants a total?
They find only one term instead of adding the terms. If the question asks for the total, use Sₙ.
One term is not total
For 5 + 10 + 20 + 40 + ... what are a and r?
a = 5 and r = 2.
Read first term + multiplier
Why is a geometric series useful in applications?
It adds repeated growth amounts together, so it is useful when the question wants a running total, not just the latest value.
Total growth
If r = 1, can you use Sₙ = a(1 − rⁿ)/(1 − r)?
No. The denominator becomes 0. If r = 1, every term is the same, so Sₙ = n × a.
Special case
1.3.38 cards
How do you recognise a geometric growth or decay situation?
Look for the same percentage change each period. Constant percentage change means geometric.
Percentage each step
What multiplier do you use for p% growth?
r = 1 + p/100
Growth multiplier
What multiplier do you use for p% decay?
r = 1 − p/100
Decay multiplier
For a 15% yearly loss in value, what is r?
r = 1 − 15/100 = 0.85
Loss -> subtract
What should the exponent on r represent in a growth/decay model?
The number of periods that have passed. It is the number of times you multiply by r.
Count the periods
If a calculator gives 6.85 years for “first exceeds” or “first drops below”, how do you round?
Round up. You need the first whole period where the threshold has actually been crossed.
Threshold question
Why is “adds 5% of the original value each year” not geometric?
Because the amount added is fixed each year. It is arithmetic, not geometric.
Original value trap
How should a final answer in a growth/decay problem be written?
Give the value with sensible rounding, units, and a short sentence in context.
Finish in context
1.3.48 cards
What condition must hold for S∞ to exist?
|r| < 1 — the terms must be getting smaller toward zero.
Think: what happens to terms if r = 2 vs r = 0.5?
Write the Sum to Infinity formula.
S∞ = u₁ ÷ (1 − r). Only valid when |r| < 1.
The denominator is (1 − r), not r.
Does S∞ exist for: 3 + 6 + 12 + 24 + ... ?
No. r = 6 ÷ 3 = 2. |r| = 2 ≥ 1, so S∞ does not exist.
Find r first, then check |r|.
Does S∞ exist for: 10 + 5 + 2.5 + ... ? If yes, find it.
r = 0.5. |r| = 0.5 < 1 ✓. S∞ = 10 ÷ (1 − 0.5) = 20.
Check |r| < 1 first, then apply the formula.
S∞ = 30 and r = 0.4. Find u₁.
u₁ = S∞ × (1 − r) = 30 × (1 − 0.4) = 30 × 0.6 = 18.
Rearrange: multiply both sides by (1 − r).
u₁ = 12 and S∞ = 20. Find r.
1 − r = u₁ ÷ S∞ = 12 ÷ 20 = 0.6, so r = 0.4.
Sub into S∞ = u₁ ÷ (1 − r) and isolate r.
r = −0.6. Does S∞ exist? Explain.
Yes. |r| = |−0.6| = 0.6 < 1 ✓. Negative r is fine — |r| strips the sign.
|r| means absolute value. Strip the minus.
Exam rule: what must you write before calculating S∞?
State: |r| < 1 ✓. IB mark schemes award this step — you earn the method mark even if the final answer is wrong.
Never skip the check. It is worth marks on its own.
Topic 1.3 study notes
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