Analysis only, and the shortest sub-topic in Topic 1: one formula and one condition. The condition is the content.
Go to the r = 3 view and leave it up. The dots climb out of the top of the frame while the formula's line sits below the axis at −1. A sum of positive numbers, and the answer is under every one of them.
Then back to r = 0.6, where the dots approach the line and stop short of it. Ask whether the partial sums ever reach 5. They do not, and 5 is still the right answer, because an infinite sum means the value the partial sums approach. That sentence is the definition and it is worth writing on the board.
The r = −0.6 view is the one that pays off later: it converges, and the dots close in from both sides. A class that has seen it will not test r > 0 when the question says |r| < 1.
| Question | Answer |
|---|---|
| 1. a = 2, r = 0.6, sum to infinity | 2/0.4 = 5. |
| 2. The tenth partial sum | 2(1 − 0.610)/0.4 = 4.97. |
| 3. First term 5, sum 20, find r | 1 − r = 0.25, so r = 0.75. |
| 4. a = 2, r = 3 in the formula | −1. |
| 5. What that −1 means | B. Nothing: |r| is not below 1, so there is no sum and the formula does not apply. |
Question 4 asks only what comes out of the arithmetic, and question 5 asks what it means. Splitting them is deliberate: a student who gets −1 and writes "so the sum is −1" has done the first correctly and the second not at all, and the two-part structure makes that visible to them rather than to you.
1 markThe condition |r| < 1, checked or stated.
1 markThe formula, with a and r in the right places.
1 markThe answer, exact where it can be.
On "for which values of r does this converge" the inequality IS the answer, and −1 < r < 1 earns it while r < 1 does not. Mark a set of those strictly once and the habit sticks.
| They wrote | What happened |
|---|---|
| 3.33 | On question 1, divided by r rather than by 1 − r. The commonest arithmetic slip here. |
| 1.25 | Used r = −0.6. They have the method and the wrong sign, probably from the figure. |
| 0.2 | The formula upside down, (1 − r)/a. |
| 4.97 on question 1 | Gave the tenth partial sum. Worth a sentence: the partial sums never arrive, and the limit is the answer. |
| 5 on question 2 | The reverse. They have not noticed the question changed. |
| 0.0202 | On question 2, gave the tenth TERM rather than the sum of ten. |
| 0.25 | On question 3, stopped at 1 − r. One step from correct. |
| 4 | On question 3, gave 20/5, which is 1/(1 − r). Two steps from correct and a good one to talk through. |
| 1 on question 4 | Treated 1 − 3 as 2. The sign is the whole point of the question, so this one costs the lesson rather than a mark. |
| "The sum is −1" on question 5 | The misconception the page is for. They trust the formula over the series. Ask for the first four partial sums. |
"Is 0.999 recurring really 1?" Yes, exactly, and this is the sub-topic that proves it: a = 0.9, r = 0.1, so the sum is 0.9/0.9 = 1. Not nearly 1, not a different number that rounds to 1. The two notations name the same point on the line. Expect an argument and welcome it; the series is the answer to every objection.
"What if r = 1?" The denominator is zero and the formula has nothing to return, which is honest: the series is 2 + 2 + 2 + … and the partial sums are 2n. Worth doing because the division by zero is the one case where the arithmetic refuses rather than lying.
"And r = −1?" The formula returns 1, and the partial sums alternate between 2 and 0 for ever. 1 is their average and is not a value the series ever takes. This is the sharpest example of a formula producing a plausible number outside its domain, and it is worth thirty seconds with a strong class.
"How do I know whether to use Sn or S∞?" Read the question for a number of terms. If there is one, it is Sn; if the question says "to infinity", "in the long run" or "eventually", it is S∞. Context questions about bouncing balls and repayments nearly always want the second, and the words to look for are the ones above.
| Stage | What to do |
|---|---|
| Demonstrate | Tabulate the partial sums for r = 0.6 to n = 12, then change the 0.6 to a 3 and run the same column. 4.97 against 59048. Do not explain the second column; ask what the sum to infinity is and let someone say that there is not one. |
| Where they stick | Setting the table range. On the Casio it is F5 SET and then F6 TABL, and a table that opens empty is almost always a start value above the end value. On the Nspire, Generate Sequence lives under Data and not under Actions, which is where people look. |
| The check | For 0 < r < 1 the sum to infinity sits above every partial sum and above the first term: 5 > 4.97 > 2 here. Do not state it more widely than that, because at r = −0.6 the limit is 1.25 while S₁ is 2 and S₃ is 1.52, both above it. The version that always holds is that S∞ lies between any two consecutive partial sums, and for a negative r the partial sums close in on it from alternate sides. |
Analysis Paper 1 has no calculator, and these answers are usually exact anyway: 2/0.4 is 5 and 0.9/0.9 is 1. The table is a Paper 2 tool and a teaching tool, not a method.
| Step | What |
|---|---|
| 1 | Fill in the partial sums for a = 2, r = 0.6 together, to n = 5. Ask where they are heading. |
| 2 | Derive S∞ from Sn by killing rn. Name the condition as you do it. |
| 3 | Figure at r = 0.6, then r = 3. Say nothing on the second. |
| 4 | Unpack why: rn does not die, so there was nothing to drop. |
| 5 | r = −0.6, which converges, and r = 1, which divides by zero. |
| 6 | Backwards from a sum, with the |r| < 1 check done on the answer. |
| 7 | 0.999 recurring, as the closing five minutes. |
Do not say "the series gets infinitely close to 5, so we call it 5". It sounds like a convention rather than a fact and it invites the reply that it is not really 5. Say that 5 is the value the partial sums approach and that no other number has that property, which is both true and the actual definition.
Do not say "check that r is less than 1". That passes r = −4, which diverges, and it is a sentence students write down verbatim. Say the modulus every time, and write |r| < 1 rather than r < 1 on the board, including in the margin of their books.