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Flip to reveal answersWhat condition must hold for S∞ to exist?
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All 8 Flashcards — Sum to Infinity
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Question
What condition must hold for S∞ to exist?
Answer
|r| < 1 — the terms must be getting smaller toward zero.
💡 Hint
Think: what happens to terms if r = 2 vs r = 0.5?
Question
Write the Sum to Infinity formula.
Answer
S∞ = u₁ ÷ (1 − r). Only valid when |r| < 1.
💡 Hint
The denominator is (1 − r), not r.
Question
Does S∞ exist for: 3 + 6 + 12 + 24 + ... ?
Answer
No. r = 6 ÷ 3 = 2. |r| = 2 ≥ 1, so S∞ does not exist.
💡 Hint
Find r first, then check |r|.
Question
Does S∞ exist for: 10 + 5 + 2.5 + ... ? If yes, find it.
Answer
r = 0.5. |r| = 0.5 < 1 ✓. S∞ = 10 ÷ (1 − 0.5) = 20.
💡 Hint
Check |r| < 1 first, then apply the formula.
Question
S∞ = 30 and r = 0.4. Find u₁.
Answer
u₁ = S∞ × (1 − r) = 30 × (1 − 0.4) = 30 × 0.6 = 18.
💡 Hint
Rearrange: multiply both sides by (1 − r).
Question
u₁ = 12 and S∞ = 20. Find r.
Answer
1 − r = u₁ ÷ S∞ = 12 ÷ 20 = 0.6, so r = 0.4.
💡 Hint
Sub into S∞ = u₁ ÷ (1 − r) and isolate r.
Question
r = −0.6. Does S∞ exist? Explain.
Answer
Yes. |r| = |−0.6| = 0.6 < 1 ✓. Negative r is fine — |r| strips the sign.
💡 Hint
|r| means absolute value. Strip the minus.
Question
Exam rule: what must you write before calculating S∞?
Answer
State: |r| < 1 ✓. IB mark schemes award this step — you earn the method mark even if the final answer is wrong.
💡 Hint
Never skip the check. It is worth marks on its own.
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Full study notes for Sum to Infinity
Topic 1.3 hub
Geometric sequences
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