Progressions puzzles

Two facts about an AP give two equations in a and d. If t₁₁ = 38 and t₁₆ = 73, the 5 jumps between them add 35, so d = 7 and a = 38 – 10 × 7 = –32. Then t₃₁ = –32 + 30 × 7 = 178.

To count multiples, list them as an AP. Three-digit multiples of 7 run from 105 to 994: (994 – 105)/7 + 1 = 128. Don't forget the + 1: counting the jumps misses the first term.

For a GP, divide terms to remove a. If the 5th term is 4 times the 3rd, then r² = 4, so r = 2 or –2.

Two GP tricks:

• write three GP terms as a/r, a, ar: their product is a³, which pins down a at once;
• equally spaced terms of a GP form a GP: the 4th, 10th and 16th terms are ar³, ar⁹, ar¹⁵, with common ratio r⁶.

Some recursions hide simpler ones.

• Pₙ = P₁ + … + Pₙ₋₁ + 1 turns into Pₙ = 2Pₙ₋₁ (powers of 2);
• Wₙ = W₁ + … + Wₙ₋₂ + 2 turns into the Virahānka rule Wₙ = Wₙ₋₁ + Wₙ₋₂.