Arithmetic progressions

Fig. 8.3 grows by 4 squares a stage: 1, 5, 9, 13, 17, 21, …. Stage n has 1 + (n – 1) × 4 = 4n – 3 squares.

A sequence with a constant difference between neighbouring terms is an arithmetic progression (AP). The constant is the common difference d. It can be negative: 11, 7, 3, –1, –5, … has d = –4.

The nth term: tₙ = a + (n – 1)d. Why n – 1? To reach the 10th term from the first you take 9 jumps of d, not 10. Each term has one fewer d than its position. The recursive rule is t₁ = a, tₙ = tₙ₋₁ + d.

APs are straight lines. Plot (n, tₙ) for an AP and the points lie on a straight line, just like the linear patterns of Chapter 2. The common difference is the slope.

Example 5. A taxi charges ₹200 to book plus ₹40 a km. After 1, 2, 3 km the fare is 240, 280, 320, …: an AP with nth term 200 + 40n. So 10 km costs ₹600.

Not every sequence is an AP. The triangular numbers, the squares and the Virahānka sequence all have changing differences.