Arithmetic Progressions
Arithmetic Progressions tests the nth term, sum of n terms, and identifying a progression. Questions often combine a term with a sum or ask how many terms are needed.
Core rules
An AP has first term a and common difference d.
- nth term: a_n=a+(n-1)d.
- Sum of first n terms: S_n=n/2[2a+(n-1)d].
- If last term l is known: S_n=n/2(a+l).
- nth term from the end of an AP with last term l: l-(n-1)d.
- Three terms in AP can be taken as a-d, a, a+d.
Worked example
For 7, 11, 15, …, find the 20th term and the sum of the first 20 terms. a=7, d=4. a_20=7+(20-1)4=7+76=83. S_20=20/2(7+83)=10×90=900.
Traps
Use n-1, not n, in the nth term. For a decreasing AP, d is negative; subtract correctly. When finding the number of terms, n must be a positive integer-discard fractional or negative results. The sum formula with l needs the last term, not the nth term from the beginning unless they coincide. Do not assume a sequence is AP from two terms; check a constant difference across at least three.