Sequences » Geometric progression

In a geometric sequence the factors between adjacent terms are the same.
For a geometric sequence with factor r and initial term t₀ is:

The direct formula: t_n = t₀ · rⁿ.

De recursive formula: t_n = r · t_n – 1 with initial term t₀.

The sum of a geometric sequence can be calculated with:
sum = first term · (1 – factor^{number of terms})1 – factor = t₀(1 – rⁿ)1 – r

Note: The factor r can also be called the 'ratio', hence the r.

Example
Given is the following sequence: 20 + 30 + 45 + ... + 227.8125.

a.	Give the direct formula.
b.	Give the recursive formula.
c.	Calculate the sum.

Answer

a.	For the direct formula t0 = 20 and r = 1.5. Therefore the direct formula is: tn = 20 · 1.5n.
b.	The recursive formula for this t0 and r is: tn = 1.5 · tn – 1.
c.	To calculate the sum, we first need to know how many terms are in the sequence. The number of terms in the sequence is equal to the number of the last term +1, as you start counting at 0. To calculate the number of the last term, we need to solve the following equation:  20 · 1.5n  = 227.8125  1.5n  = 227.8125 : 20  n  = log(227.8125 : 20)log(1.5) = 6. The number of the last term is 6, so the total number of terms is 7. The sum of this sequence is then: sum = 20 · (1 – 1.57)1 – 1,5 = 643.4375