Geometric: Explicit Models

Geometric

Explicit Models

The following steps lead to an important discovery. Remember, when finding the next term in a geometric sequence, multiply by the common ratio, r.

First Term: a ₁

Second Term: a ₂ = a₁?r

Third Term: a ₃ = a₂?r

= (a₁?r)r = a₁r²

Fourth Term: a ₄ = a₃?r

= (a₁r²) r = a₁r³

* Notice that in each term, the exponent on r is always one less than the position of the term: a₄= a₁r^(4-1) = a₁r³ .

N^th Term: a_n = a₁r^n-1

The formula above is also known as the explicit formula and is used to determine the n^th term when the previous term is unknown. When simplified, this formula becomes: and is exponential.