Arithmetic Progressions
Arithmetic Progressions tests the common difference, nth term, sum of terms, and real-life sequences. An AP has a constant difference d between consecutive terms.
Core rules
- nth term: an = a + (n-1)d, where a is the first term.
- Sum of first n terms: Sn = n/2[2a + (n-1)d]. If the last term l is known, Sn = n/2(a+l).
- The terms between two given terms are arithmetic means; for one mean between a and b, it is (a+b)/2.
- If a sequence has a constant second difference, it is not necessarily an AP; check consecutive differences first.
- The number of terms is found from an = l, not by assuming n equals the last term.
Worked example
Find the sum of the first 20 terms of 3, 7, 11, ... Here a=3 and d=4. S20 = 20/2[2(3)+(20-1)4] = 10[6+76] = 10×82 = 820. The 20th term is 3+19×4=79; using Sn = n/2(a+l) gives 10(3+79)=820, confirming the result.
Traps
The formula uses n-1, not n. Distinguish the nth term from the sum of n terms. When counting terms from a to l, solve a+(n-1)d=l carefully. A negative d is valid and means the AP decreases. Do not average only the first and last terms unless the sequence is an AP.