subject: Geometric Sequence Series [print this page] In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 1/2 + 1/4 + 1/8 + 1/18+ is geometric, because each term except the first can be obtained by multiplying the previous term by 1/2.
(Source: from wikipedia).
The two simplest sequences to work with are arithmetic and geometric sequences. An arithmetic sequence goes from one term to the next by always adding (or subtracting) the same value. For instance, 2, 5, 8, 11, 14,... and 7, 3, 1, 5,... are arithmetic, since you add 3 and subtract 4, respectively, at each step. A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value. So 1, 2, 4, 8, 16,... and 81, 27, 9, 3, 1, 1/3,... are geometric, since you multiply by 2 and divide by 3, respectively, at each step.
The number added (or subtracted) at each stage of an arithmetic sequence is called the "common difference" d, because if you subtract (find the difference of) successive terms, you'll always get this common value. The number multiplied (or divided) at each stage of a geometric sequence is called the "common ratio" r, because if you divide (find the ratio of) successive terms, you'll always get this common value.
General term of a Geometric sequence series:
Let us consider a Geometric series with first non-zero term a and common ratio r. Write a few terms of it. The second term is obtained by multiplying a with r, thus a2 = ar. Similarly, third term is obtained by multiplying a2 by r. Thus, a3 = a2r = ar2, and so on.
We write below these and few more terms.
The general form is an=ar^n-1
1st term = a1 = a = ar^11
2nd term = a2 = ar = ar^21
3rd term = a3 = ar2 = ar^31
4th term = a4 = ar3 = ar^41
5th term = a5 = ar4 = ar^51
a = the first term, r = the common ratio, l = the last term,
n = the numbers of terms, Sn = the sum of n terms.
Students can also utilize the cbse syllabus class ix available online for the reference.
Sum to n terms of a Geometric Sequence series:
Let the first term of a geometric series be a and the common ratio be r. Let us denote by Sn. It is the sum to first n terms of Geometric series. Then
