The other face of the sum
A savings box gets ₹7 in month 1. Each month it gets ₹4 more than the month before. How much has it got after 12 months? No one has stated the last month's amount.
In 12 months the box has got ₹348 in all.
The last term is l = a + (n − 1)d. Put this l into Sₙ = n/2 (a + l). Then a + l = a + a + (n − 1)d = 2a + (n − 1)d. So the total is written with a and d only.
Rule: for a list with the same step each time, Sₙ = n/2 (2a + (n − 1)d). Here a is the first term and d is the step. The letter n is how many terms.
Nothing new was added. The last term is just written using a and d. We call this the other face of the sum.
| List | a | d | n | 2a + (n−1)d | Sₙ |
|---|---|---|---|---|---|
| Savings box | 7 | 4 | 12 | 14 + 44 = 58 | 6 × 58 = 348 |
| 3, 7, 11, … | 3 | 4 | 10 | 6 + 36 = 42 | 5 × 42 = 210 |
The savings box has got ₹348 after 12 months. The list 3, 7, 11, … totals 210 over its first 10 terms.
Watch out!
Write (n − 1)d, not nd. The first term has no step yet. Twelve terms use eleven steps.
Slide a, d and n. Watch the bars and the total.
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