Arithmetic Sequences

Arithmetic Sequences: A sequence in which the difference between each term and the preceding term is always constant.

The Explicit Formula: an = a1 + (n – 1)d An explicit formula, for arithmetic sequences, allows you to find a term within the sequence based on the position in the sequence the term is in. The Recursive Formula: The recursive formula, for arithmetic sequences, allows you to find a term by using the preceding terms in the sequence.

EXAMPLE: 
The First term, U₁, and the common ratio, r, of a geometric sequence are given. Find the sixth term and the recursive and explicit formulas for the nth term

U₁ = 1, r = -2
Sixth Term: U₆ = 1(-2)⁵---U₆ = -32

Recursive Form:
U₁=1

U₂=(-2)1=-2

U₃=(-2)(-2)=4

U₄=(4)(-2)=8

U₅=(-8)(-2)=16

U₆=(16)(-2)=-32

Explicit Form:

U₁=1

U₂=(1)(-2)=-2

U₃=(1)(-2)²=4

U₄=(1)(-2)³=-8

U₅=(1)(-2)⁴=16

U₆=(1)(-2)⁵=-32