Arithmetic Progressions
20 marks
An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number to the preceding term (except the first term).
This fixed number is the common difference d:
d = a₂ − a₁ = a₃ − a₂
in general d = aₖ₊₁ − aₖ
d can be positive, negative or zero. An AP with a last term is finite; otherwise infinite.
a = first term · d = common difference · n = number of terms · aₙ = nth term · l = last term · Sₙ = sum of first n terms
★ Formulas
aₙ = a + (n − 1)d
nth term from the end: l − (n − 1)d
Sₙ = n2 [2a + (n − 1)d]
Sₙ = n2 (a + l) ← use when the last term l is known
aₙ = Sₙ − Sₙ₋₁
and a₁ = S₁
If a, b, c are in AP:
2b = a + c
b = a + c2 is the arithmetic mean
3 terms: a − d, a, a + d
4 terms: a − 3d, a − d, a + d, a + 3d
(sum becomes easy — the d’s cancel)
Special sums
- If aₙ = pn + q (linear in n), it is an AP with d = p.
- If Sₙ = An² + Bn (no constant), it is an AP with d = 2A.
- Middle term of n terms: n odd → (n + 12)ᵗʰ; n even → (n2)ᵗʰ & (n2 + 1)ᵗʰ.
- “Is 301 a term of the AP?” → solve aₙ = 301; n must be a positive integer.
- “First negative term” → solve aₙ < 0 and take the smallest such n.
- d = a₂ − a₁, not a₁ − a₂.
- n (number of terms) can never be negative, zero or a fraction.
- Number of terms between two values: “multiples of 7 between 1 and 100” → find first & last multiple, then use aₙ.
- Adding/multiplying every term by a constant gives another AP (with d unchanged / multiplied).