There is absolutely much to know about series and sequences that may be effective for dealing with various events taking place in regular lives and other areas of professional working.
Geometric Progression
In a sequence of terms, if each term is observed to be a constant multiple of the preceding term, the sequence is then addressed as a geometric progression (or GP). Also, the considerable constant multiplier for the elements in the series is referred as the common ratio. Recall that in arithmetic progression, the difference between the nth and (n – 1)th term is supposed to be a constant which is referred as the common difference of that arithmetic progression. You know that an arithmetic progression exists in the form:
x, x + d, x + 2d, x + 3d……x + (n – 1)d
Now consider a situation where the ratio of nth term to (n – 1)th term in a sequence would be a constant. Consider the following sequence for instance,
2, 4, 8, 16, 32, ………
Here, you may observe that: $\frac{4}{2}=\frac{8}{4}=\frac{16}{8}=\frac{32}{16}=2$
Now, consider another series: $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$ ………..
$\frac{\frac{1}{2}}{1}= \frac{\frac{1}{4}}{\frac{1}{2}}=\frac{\frac{1}{8}}{\frac{1}{4}}=\frac{\frac{1}{16}}{\frac{1}{8}}=\frac{1}{2}$
In the examples given above, the ratio is observed to be constant. This type of sequence is referred to as geometric progressions and abbreviated as (G.P.). If the sequence:
p1, p2, p3, p4, …….pn…..is a G.P then in that case, $\frac{p_{k}+1}{p_{k}}=r$
Also, here [k > 1]; In this situation, r is supposed to be a constant and this is further known as a common ratio, considering the fact that none of the terms in this sequence is zero.
General Term in a Geometric Progression
We already know how to make out the nth term of an arithmetic progression. This is with the equation or formula: $x_{n}=x+(n-1)d$
Considering a similar situation for geometric progression, let p be the first term and r be the common ratio. Then in that case,
The second element or term would be: $p_{2}=p\times r=pr$;
Also, the third term would be: $p_{3}=p_{2}\times r=pr^{2}$;
In the same pattern that would follow, the nth term for the sequence would be calculated as: $p_{n}=pr^{n-1}$.
You may further calculate the sum of nth term of the GP based on the fact that it is a finite or infinite progression.
A finite element for a GP could be written as: $p, pr, pr^{2},pr^{3},pr^{4},…..pr^{n-1}$.
Also, the elements for an infinite GP could be written as: $p, pr, pr^{2},pr^{3},pr^{4},….. pr^{n-1}…..$
The extension of dots represents that the progression would be taken forward till infinity.
