Revise: Sequences and Series

aₙ = a₁ + (n−1)d.

a₁=5,d=3 → a₁₀=32.

Sₙ = n/2·(a₁+aₙ).

1+2+...+10 = 55.

aₙ = a₁·rⁿ⁻¹.

a₁=3,r=2 → a₆=96.

Sₙ = a₁(1−rⁿ)/(1−r).

a₁=2,r=3,n=5 → S₅=242.

S∞ = a₁/(1−r), only if |r|<1.

8+4+2+1+... = 16.

Sequence = list of terms. Series = sum of terms.

aₙ=2n+1 → a₁,a₂,a₃=3,5,7, S₃=15.

d = common difference, added each step.

a₁=4,a₅=20 → d=4.

Pair first+last: n/2 pairs, each summing to a₁+aₙ.

a₁=3,d=5,n=20 → S₂₀=1010.

r = common ratio, multiplied each step.

a₁=2,a₄=54 → r=3.

Proof: subtract rSₙ from Sₙ, most terms cancel.

a₁=1,r=2,n=4 → S₄=15.

As n→∞, rⁿ→0 when |r|<1, so Sₙ approaches a₁/(1−r).

0.333... = 1/3 via a₁=0.3,r=0.1.