Sequences & Series

Sequences & series with a common difference — nth term, sums & sigma notation. ✎

① Arithmetic Sequences

Each term is the previous one plus a fixed common difference d.

d = u n+1 − u n

Check it's arithmetic: the difference between any two consecutive terms is the same.

② The nth Term ✦ booklet

u n = u₁ + (n − 1)d

u₁ = first term · d = common difference.

It's just u₁ plus (n−1) jumps of size d — that's why it's n−1, not n.

Plots as a straight line (gradient d) — arithmetic = linear.

e.g. 4, 7, 10, 13, … → u₁=4, d=3 u n = 4 + (n−1)·3 = 3n + 1

③ Arithmetic Series (sum) ✦ booklet

A series is the sum of the first n terms. Two equivalent formulas:

S n = n / 2 [ 2u₁ + (n − 1)d ]

— use when you know d —

— use when you know the last term u n —

④ Sigma Notation Σ

Σ (sigma) is shorthand for "add them all up":

n=1

(2n + 1)

3 + 5 + 7 + 9

Bottom (n=1) = where to start · top (4) = where to stop.

Substitute n = 1, 2, 3, 4 into the rule, then add.

Number of terms = top − bottom + 1.

★ Worked example ✎

An arithmetic sequence has u₁ = 5 and d = 3.

Find u₁₀: u₁₀ = 5 + (10−1)·3 = 5 + 27 = 32

Exam tip!

Always find u₁ and d first — every arithmetic question unlocks from those two numbers.

Geometric

Sequences & series with a common ratio — nth term, sums & the sum to infinity. ✎

① Geometric Sequences

Each term is the previous one multiplied by a fixed common ratio r.

r = u n+1 / u n

u n = u₁ · r n−1

u₁ = first term · r = common ratio.

Multiply by r a total of (n−1) times to reach the nth term.

Grows exponentially (curved), not linearly.

e.g. 3, 6, 12, 24, … → u₁=3, r=2 u n = 3 · 2 n−1

③ Geometric Series (sum) ✦ booklet

Sum of the first n terms (r ≠ 1) — two equivalent forms:

S n = u₁(r n − 1) / r − 1

— best when r > 1 —

S n = u₁(1 − r n ) / 1 − r

— best when r < 1 (keeps it positive) —

④ Sum to Infinity S∞ ✦ booklet

S ∞ = u₁ / 1 − r

only converges when |r| < 1

When terms shrink, the total approaches a finite limit. Picture ½ + ¼ + ⅛ + … filling one whole square:

⑤ Compound Interest ✦ booklet

Money growing by a fixed % is a geometric sequence:

r = annual interest rate (%) · n = years.

k = compounding periods per year (12 = monthly).

A geometric sequence has u₁ = 8 and r = ½.

Find u₄: u₄ = 8 · (½)³ = 8 · ⅛ = 1

Sum to infinity? |½| < 1 ✓ so it converges:

1 − ½

Always check |r| < 1 before using S∞ — if not, the sum diverges and has no finite value.

made for revision · IB AA SL · keep it by your desk ✦