0Geometric Progression(GP) or Geometric Sequence is sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant.
A geometric progression(GP) is given by a, ar, ar2, ar3,
where a = the first term, r = the common ratio
nth term of a geometric progression(GP):
tn=arn−1">tn=arn−1
where tn = nth term, a= the first term, r = common ratio, n = number of terms
Sum of first n terms in a Geometric Progression (GP):
Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Sn=a(rn−1)/r−1 (if r>1)Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">a(1−rn)/1−r (if r<1)
Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Sum of an infinite geometric progression(GP)
Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">S∞=a1−r (if -1 < r < 1)">S∞=a/1−r (if -1 < r < 1)
Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">where a= the first term, r = common ratio
Geometric MeanIf three non-zero numbers a, b, c are in GP, b is the Geometric Mean(GM) between a and c. In this case, b=ac">b=√ac
b=ac">The Geometric Mean(GM) between two numbers a and b = ab">√ab
(Note that if a and b are of opposite sign, their GM is not defined.)
Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">Additional Notes on GP
To solve most of the problems related to GP, the terms of the GP can be conveniently taken as:
3 terms: ar">Sn={a(rn−1)r−1 (if r>1)a(1−rn)1−r (if r<1)">a/r, a, ar
5 terms: ar2">ar2, ar">ar, a, ar, ar2.
