The nth term

Write an AP out using a and d:

Notice: the number of d's is always one less than the term number. So the nth term is:

aₙ = a + (n – 1)d

Want the 100th term? Don't add d ninety-nine times. Just work out a + 99d.

Reena's salary: a = 8000, d = 500. In year 5 it is 8000 + 4 × 500 = ₹10,000.

'Which term is this number?' Call it the nth term and solve for n. n must be a positive whole number. Is 301 a term of 5, 11, 17, …? 301 = 5 + (n – 1)6 gives n = 151/3. That is not a whole number, so 301 is not a term.

'Find the AP' means find a and d. You need two facts, so you get two equations. If a₃ = 5 and a₇ = 9, then a + 2d = 5 and a + 6d = 9. Subtracting gives 4d = 4, so d = 1 and a = 3.

How many numbers in a list? Find the first and last, then solve for n. Two-digit multiples of 3 run 12, 15, …, 99, so 99 = 12 + (n – 1)3 and n = 30.

Counting from the end: turn the AP round. The last term becomes the first, and d changes sign. In a 25-term AP, the 11th term from the end is the 15th from the start.