Using the sum formula

The sum formula Sₙ = n/2 [2a + (n – 1)d] links four things: S, a, d and n. Know any three and you can find the fourth. Combine it with aₙ = a + (n – 1)d when the question gives terms.

First, say in words what you know. 'The 3rd year made 600 TV sets' is a₃ = 600. 'Total in the first 7 years' is S₇.

• TV sets: a₃ = 600 and a₇ = 700. Two equations give d = 25 and a = 550. Then S₇ = 7/2 (1100 + 150) = 4375 sets.
• Prizes: seven prizes add to ₹700, each ₹20 less than the one before. 700 = 7/2 [2a + 6(–20)] gives the first prize a = ₹160.

Finding n often gives a quadratic, and you must check both roots. A pile of logs has 20 in the bottom row, 19 in the next, 18 in the next, and so on. How many rows make 200 logs?

200 = n/2 [40 – (n – 1)] gives n² – 41n + 400 = 0, so n = 16 or 25.

Try 25: the top row would have 20 – 24 = –4 logs, which is impossible. So 16 rows, with 5 logs in the top row. Always say why you reject an answer.

The middle of three. If a, b, c are in AP, then b – a = c – b, so b = (a + c)/2. This is called the arithmetic mean of a and c. Example: 8 is the arithmetic mean of 5 and 11.