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: tn = t0 · rn.
De recursive formula: tn = r · tn – 1 with initial term t0.
The sum of a geometric sequence can be calculated with:
sum = first term · (1 – factornumber of terms)1 – factor = t0(1 – rn)1 – r
Note: The factor r can also be called the 'ratio', hence the r.
Given is the following sequence: 20 + 30 + 45 + ... + 227.8125.
|a.||Give the direct formula.|
|b.||Give the recursive formula.|
|c.||Calculate the sum.|
|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:
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