Arithmetic Progressions — Revision Notes

⚡ 30-Second Revision

Revision checked 17 October 2026.

d = next − current; d may be negative or zero.
a_n = a + (n − 1)d.
S_n = n(first + last)/2.
Membership needs an integer index.
Count inclusive endpoints; distinguish term from total.

2-Minute Revision

From fourth term 19 and ninth term 44, five steps contribute 25, so d = 5 and a = 4. A row sequence starting at 14 and adding 3 has twelfth term 47 and twelve-row total 366. Equal percentage growth is usually not an AP. For interval questions, choose admissible first and last terms before counting. A constant sequence needs separate treatment when a formula would divide by zero.

5-Minute Revision

Original self-test: A sequence begins 12, 18, 24. Find its fifteenth term: 12 + 14 × 6 = 96. Find its first fifteen-term sum: 15(12 + 96)/2 = 810. Test whether 95 belongs: (95 − 12)/6 + 1 is not an integer. Now reverse the question: if the fifth term is 36 and tenth is 66, the five-step increase gives d = 6 and a = 12. Each result checks a different operation; do not substitute a sum when the stem requests one term.

Vyyuha Quick Recall

STEP: Spot the constant difference; Track the index; Endpoints for sums; Positive integer check for membership.