Series (Number & Letter)

high importance~4 Q in Tier 16 formulas⚡ 10 shortcuts6 subtopics

Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.

Number series

AP / ladder

a_n = a_1 + (n-1)d

d = common difference from the ladder.

Recurrence xk plus r

a_{n+1} = k \cdot a_n \pm r

Test k = 2, 3 first with r = ±1, ±2.

n-squared family

a_n = n^2 \pm k \text{ or } n^2 + n

Second difference 2 ⇒ n²; second difference 2k ⇒ k·n².

Geometric

a_n = a_1 \cdot r^{\,n-1}

Constant ratio r; check ×2, ×3, ×1.5.

Slot steps

s_i = Q(L_{i+1}) - Q(L_i) \pmod{26}

Steps per slot decide the rule; wrap Y→B counts as +3.

Two-track cluster

Q(\text{first}_i) = a + d_1 i,\quad Q(\text{second}_i) = b + d_2 i

Tracks are independent; solve both.