Why u1 + nd survives all the way into examinations, and the one check that kills it.
Run it at n = 1 first, and only n = 1.
The correct formula returns 5, the first term. The off-by-one returns 8, the second. That is the entire demonstration and it takes ten seconds.
Then step out to n = 12 and show the gap is still exactly 3. It never grows and never closes, which is why the wrong answer always looks plausible: it is a real term of the sequence, just the next one.
Put n = 1 back in. If the formula does not return the first term, it is wrong. It costs five seconds and it catches the error every time.
Make them do it on every sequence question for a fortnight and it becomes a habit that survives the examination. It also transfers directly to 1.3, where the same minus one sits in the power.
| 1. u₂₀ | 5 + 19(3) = 62. |
| 2. S₂₀ | (20/2)(5 + 62) = 670. |
| 3. d from u₃ = 11, u₇ = 23 | 12 over 4 steps = 3. |
1 markStating u₁ and d explicitly before using either formula.
1 markThe (n − 1), verified at n = 1.
1 markChoosing the shorter sum formula when the last term is already known.
| They give | What it means |
|---|---|
| 65 (Q1) | Twenty steps instead of nineteen. The missing minus one, and the answer is u₂₁. |
| 62 (Q2) | Gave the term when the sum was asked for. Worth separating from a method error; it is a reading slip. |
| 1340 (Q2) | Forgot to halve. |
| 1.714 or 2.4 (Q3) | Divided by 7 or by 5 rather than by the 4 steps between u₃ and u₇. Make them count the steps on their fingers; it is not beneath anyone. |
| 12 (Q3) | Gave the total increase, not the per-step difference. |
"Which sum formula do I use?" If you know the last term, use n/2(first + last). If you do not, use the other one. They are the same formula with uₙ substituted.
"Can d be negative?" Yes, and then the sequence decreases and the sum can turn negative. Worth one example so it is not a surprise.
"What if n comes out as a decimal?" Then the value is not a term of the sequence, and saying so is the answer. It is not a sign to round.
| What is happening | |
|---|---|
| 1 | n = 1 on both formulas. Let the gap be the lesson. |
| 2 | Both formulas, with the n = 1 check written in every time. |
| 3 | The three questions. |
| 4 | Backwards questions: find d, find n, find u₁ from two terms. |
| 5 | A context where the model is not perfectly arithmetic, and what to say about it. |
Do not write the formula without the brackets round n − 1. They will drop them.
Do not skip the n = 1 check once they can do it. It is the habit, not the arithmetic, that is being taught.
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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