The big picture
Here is the track, one idea to a line.
| Idea | Remember | Example |
|---|---|---|
| Every end pair | The step cancels; every end pair totals a + l. | Sₙ = n/2 (a + l) |
| The other face | Open l into a and d: Sₙ = n/2 (2a + (n − 1)d). | Savings box, ₹348 |
| Work it backwards | With a and d known, unknown n gives a quadratic. That needs d not 0. Keep only a whole count of terms, at least 1. | 5, 8, 11, … → 75, n = 6 |
| Terms from the sums | a₁ = S₁, since nothing comes before the first term. Then aₙ = Sₙ − Sₙ₋₁ for n = 2, 3, 4, and so on. | Sₙ = 2n² + 3n → 5, 9, 13, 17 |
| Two sums name it | Two totals, for different counts of terms, give two linear equations. | S₅ = 30, S₁₃ = 182 → a = 2, d = 2 |
| Stacks and rows | Read a, d, then term-or-total. The story can show that an answer is wrong. | Log pile, 120 logs |
| When it slides | A negative step is a real step. The slide ends where a term reaches zero. | Day 16, tour stops |
| Which question | Ask two things. What is given? What is missing? | Rows 5 to 15: 555 − 82 |
| Is it a term | Put the number in aₙ = a + (n − 1)d. A term needs n to be a whole number, at least 1. | 104 in 4, 9, 14, …: n = 21 |
| Two facts | Write each fact as a + (n − 1)d. Subtract to get d. Then find a. | 4th term 14, 9th term 34: 2, 6, 10, … |
Chapter 5 of your Mathematics book is Arithmetic Progressions. You could name a term. Now you can total a whole list, and work backwards too.
The track began with the hall: 295 chairs, five pairs of 59. That pairing is inside both formulas above.