Geometric Series is simply multiplying every number or member of the series by the same number. This number is known as the constant ratio. In a G.P. the ratio of any two consecutive numbers is the same number which we call it as the constant ratio.
It is denoted by the letter ‘r’ so, if we have the G.P. suppose a1, a2, a3, …, an, the ratio of any two consecutive numbers within the series, will be the same. Thus, for the series presented above, we must have –
a3 / a2 = r,
where ‘r’ is known as the common ratio. In other words, if you know that ‘r’ and the first term, you can produce the entire geometric series or say, geometric progression.
Now let us solve some of the examples –
Example – 1: In the Geometric progression or G.P., r = 2 and a = 1. Find out the tenth term.
Solution – In a geometric series, every term is just multiplied by the common ratio r. For the second term, the first term is multiplied by r. We also get the third term just by multiplying the first term r2.
In the same way, we will get the fourth term just by multiplying the first term with r3 and so on. After knowing this, the above example will become very easy. Thus, the first term is 1, we need to multiply it by 29 to get the tenth term which is 512. So, the tenth term of the G.P. = 512.
Sum of the geometric progression or geometric series
Sometimes, you will be given the series and asked to find out the sum of the first few terms or the entire geometric series. The sum is denoted by Sn,
Where n is known, as the number of terms up to which the sum is solved.
For instance, the sum of the first ten numbers or terms will be symbolized by S10. Here we will see the list of important formulas for finding out the sum of the first few terms. Suppose ‘a’ is the first term of the G.P. and the r is the common ratio, then the sum of the G.P. can be found out by applying the following formulae:
Sn = a (rn – 1) / r – 1, if r is not equal to 1 and,
Sn = an, if r is equal to 1
The sum of infinite terms of the geometric progression (G.P.) in the situation of -1 < r < 1 is given by the below formula – Sn = a / (1 –r)
Thus, the above are the three formulas which depend on the value of r. We have also seen some of the examples which clearly show the concept of geometric progression or geometric series.
Thus, this section shows you the detailed information of the geometric progression or geometric series. G.P. is the short form of the geometric progression which is also known as the geometric series.