- Nth Term of Geometric Progression (G.P.) - T_n=ar^(n-1)
- General Term of a Geometric Progression (G.P.)
- Sum of First N Terms of a Geometric Progression (G.P.) - S_n=a(r^n-1)/(r-1)
- Infinite Geometric Progression (G.P.) and Its Sum - S∞=a1−r;|r|<1S∞=a1-r;|r|<1
- Geometric Mean (G.M.)
Let us consider the following sequences: 2,4,8,16,...,
we have `a_1 = 2 , a_2/a_1 = 2 , a_3/a_2 = 2, a_4/a_3 = 2` and so on.
In above sequence the constant ratio is 2. Such sequences are called geometric sequence or geometric progression abbreviated as G.P.
1) General term of a G .P:
Let us consider a G.P. with first non-zero term ‘a’ and common ratio ‘r’. The second term is obtained by multiplying a by r, thus `a_2` = ar. Similarly, third term is obtained by multiplying `a_2` by r. Thus, `a_3 = a_2r = ar^2`, and so on.
The nth term of a G.P. is given by `a_n =ar^(n-1)`.
The series `a + ar + ar^2 + ... +` `ar^(n–1)`or `a + ar + ar^2 + ...+` `ar^(n–1) +...` are called finite or infinite geometric series, respectively.
2) Sum to n terms of a G .P.:
The first term of a G.P. be a and the common ratio be r.
`s_n =(a(1-r^n))/1-r` or `s_n = (a(r^n -1))/r-1`
3) Geometric Mean (G .M.):
The geometric mean of two positive numbers a
and b is the number `sqrt (ab)` .
`G_1, G_2,…, G_n` be n numbers between positive numbers a and b such that a,`G_1,G_2,G_3,…,G_n`, b is a G.P.
`G_n =ar^n =a (b/a)^(n/(n + 1))`