A sequence is a set of numbers is called as terms, arranged in some particular order. There are two types of sequences; they are Arithmetic sequence and geometric sequence. An arithmetic sequence is a sequence of numbers with the difference between two consecutive terms constant. The difference is called the common difference. A geometric sequence is a sequence of numbers with the ratio between two consecutive terms constant. This ratio is called the common ratio.
Geometric Sequences Definition:
In mathematics, a geometric progression is also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54 ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25 ... is a geometric sequence with common ratio 1/2. The sum of the terms of all the terms in geometric sequence is known as a geometric series.
The sum of the terms of a geometric progression is called as a geometric series. Thus, the general form of a geometric sequence is,
a, ar, ar2, ar3, ar4……
and that of a geometric series is,
a + ar + ar2 + ar3 + ar4 +…..
The nth term of a geometric sequences with initial value a and common ratio r is given by,
an = arn-1
Here, r → common ratio
n → number of term in the series
Such a geometric sequences also follows the recursive relation,
an = ran-1 for every integer n>=1
Here, r → Common ratio,
n → number of terms in the series
Generally, to check whether the given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance,
1, −3, 9, −27, 81, −243…
is a geometric sequence with common ratio −3.
Find the 4th partial sum of an = 5(2)n-1
The 4th partial sum of this series is the sum of first four terms,
a1 = 5 (2)1-1 => 5*1 => 5
a2 = 5 (2)2-1 => 5*2 => 10
a3 = 5 (2)3-1 => 5*4 => 20
a4 = 5 (2)4-1 => 5*8 => 40
Therefore, the sum of series is,
5 + 10 + 20 + 40 => 75