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All 13 Flashcards — nth term
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Question
What is a geometric sequence?
Answer
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.
Question
What is the common ratio, and how do you find it?
Answer
The constant multiplier: r = uₙ ÷ uₙ₋₁. Divide any term by the one before. Example: 12 ÷ 4 = 3.
Question
A quantity 'increases by 8% each year' vs 'increases by 8 each year' — which model, and what is the key number?
Answer
'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.
Question
Why is it rⁿ⁻¹ and not rⁿ?
Answer
You start at u₁ and multiply by r only on each step after the first — (n − 1) times. Example: u₅ = u₁ r⁴.
Question
How do you find r from two terms, e.g. u₂ = 6 and u₅ = 48?
Answer
Divide the values, then take the (steps)-th root: 48 ÷ 6 = 8 over 3 steps, so r = ∛8 = 2.
Question
When can the common ratio be negative?
Answer
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.
Question
When are three terms u₁, u₂, u₃ geometric?
Answer
When the ratios are equal: u₂/u₁ = u₃/u₂, i.e. u₂² = u₁u₃ (middle squared = product of neighbours).
Question
Find k if 4, k, 25 are geometric (k > 0).
Answer
k² = 4 × 25 = 100, so k = 10.
Question
How is geometric different from arithmetic?
Answer
Arithmetic ADDS the same d each step; geometric MULTIPLIES by the same r. 3, 6, 9 is arithmetic; 3, 6, 12 is geometric.
Question
Find u₆ for u₁ = 5 and r = 2.
Answer
u₆ = 5 × 2⁵ = 160.
Question
Why does the middle term squared equal the product of its neighbours?
Answer
Because consecutive ratios are equal: u₂/u₁ = u₃/u₂. Cross-multiplying gives u₂² = u₁u₃.
Question
Three expressions are geometric and the condition gives a quadratic. How many values of the unknown?
Answer
Up to two — solve the quadratic and report both. Use any stated condition (e.g. all terms positive) to choose between them.
Question
How do you avoid mixing up n and n − 1 in a geometric question?
Answer
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.
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Topic 1.3 hub
Geometric sequences & series
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Math AA SL exam skills
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