Why starting both sequences identically is the only version of this comparison that works.
Point out that both sequences are 3, 6 before you run it.
Drawn separately, students explain the gap away: the geometric one must have started higher, or stepped bigger. Starting both at 3, 6 removes every excuse, and by n = 10 it is 30 against 1536.
The y axis is linear on purpose. The arithmetic line lying flat along the bottom is not a flaw in the picture; that is genuinely what the comparison looks like, and a log scale would hide the only thing worth seeing.
The commonest wrong start is subtracting consecutive terms, which gives d and sends the whole question into the arithmetic formulas. Divide, and check the same ratio appears twice before committing.
Checking twice also catches sequences that are neither, which do appear in modelling questions where identifying the type is itself the mark.
| 1. u₁₀ | 3 × 2⁹ = 1536. |
| 2. S₁₀ | 3(2¹⁰ − 1)/1 = 3069. |
| 3. Sum to infinity | 4/(1 − 0.5) = 8. |
1 markFinding r by division, with the ratio confirmed on a second pair.
1 markThe n − 1 in the power.
1 markStating |r| < 1 before using the sum to infinity.
| They give | What it means |
|---|---|
| 3072 (Q1) | Used rⁿ instead of rⁿ⁻¹. The same off-by-one as 1.2, now in the exponent. |
| 30 (Q1) | Used the arithmetic sequence. They subtracted to find the step. |
| 1536 (Q2) | Gave the term, not the sum. |
| 2 (Q3) | Divided by 2 rather than by 0.5. Worth a reminder that dividing by a half doubles. |
| 7.9922 (Q3) | Gave S₁₀. Close enough to look right, which is why the exact limit matters. |
"Why does the sum stop at a number?" Because the terms shrink fast enough that the leftover is always 8rⁿ, which goes to nothing. Show the shortfall shrinking rather than asserting convergence.
"What if r is negative?" The terms alternate in sign, and the sum to infinity still exists if |r| < 1. Worth one example because it looks alarming and is not.
"Is compound interest geometric?" Yes, exactly, and that is the next lesson. Flagging it here makes 1.4 feel like a consequence rather than new material.
| What is happening | |
|---|---|
| 1 | Both sequences from the same start. Step to n = 10. |
| 2 | The two formulas, with r found by division. |
| 3 | The three questions. |
| 4 | r < 1, the shrinking shortfall, and the condition on the sum to infinity. |
| 5 | Set up 1.4: compound interest is this with r = 1 + rate. |
Do not use a logarithmic y axis to fit both sequences on. It makes exponential growth look linear, which is the opposite of the lesson.
Do not introduce the sum to infinity without the |r| < 1 condition in the same breath.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
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