Sequences & Series

In every arithmetic sequence

T1, T2 - T1, Tn - Tn-1, Tn =

T1 = a

T2 - T1 = d

Tn - Tn-1 = d

Tn = a + (n - 1)d

Sequence is arithmetic if:

Common difference is a constant (no denominator of n)

Arithmetic series

When terms of a sequence are added

To find Tn when given arithmetic series

Sn - Sn-1 = Tn

Sigma Notation

∑ = the sum of

⁴∑↓r Tr = ₁, the sum of T₁ + T₂ + T₃ + T₄

Geometric Sequences

When each term of a sequence is obtained by multiplying the previous term by a fixed amount, r = common ratio

In every geometric sequence:

T1, T2/T1, Tn, Tn+1/Tn =

T1 = a

T2/T1 = r

Tn = arⁿ⁻¹

Tn+1/Tn = r

When given product and sum of first 3 terms of a geometric sequence

Let a/r, a, a+r be first three terms, find a (ie the product). Then add the three terms and find r, then sub r and a into original terms to get sequence

To find whether a sequence is arithmetic or geometric

Common difference should equal each other

Arithmetic: T2 - T1 = T3 - T2

Geometric: T2/T1 = T3/T2