`\color{fuchsia} \(★ mathbf\ul"Geometric Progression (G . P.)")`
A sequence `color(blue)(a_1, a_2, a_3, …, a_n, … )` is called geometric progression,
if each term is non-zero and `color(red)(a_(k+1)/a_k = r)` (constant ratio ), for `color(red)(k ≥ 1.)`
`\color{red} ✍️` By letting `color(blue)(a_1 = a)`, we obtain a geometric progression, `color(red)(a, ar, ar^2, ar^3,…...,)`
where `a` is called the `color(blue)"first term"` and `r` is called the `color(blue)"common ratio"` of the G.P.
We shall use the following notations with these formulae:
`color(red)(a) = ` the first term, ` \ \ color(red)(r) = `the common ratio, ` \ \ color(red)(l) =` the last term,
`color(red)(n) =` the numbers of terms,
`color(red)(S_n) =` the sum of n terms.
`color(red)("Let us consider the following sequences:")`
`(i) color(blue)(2,4,8,16,.............,) (ii) color(blue)(1/9 , -1/ 27, 1/81 , -1/243 ..............) (iii) color(blue)(.01,.0001,.000001,..................)`
In `(i)` we have `a_1=2 , a_2/a_1=2, a_3/a_2=2, a_4/a_3=2` and so on. In `(ii)`, we observe, `a_1=1/9 , a_2/a_1=-1/3, a_3/a_2=-1/3 , a_4/a_3=-1/3` and so on.
Similarly, state how do the terms in `(iii)` progress
In `(i),` this constant ratio is `2;` in `(ii)` it is `-1/3` and in `(iii)` the constant ratio is `0.01.`
Common ratio in geometric progression `(i), (ii)` and `(iii)` above are `2, -1/3` and `0.01,` respectively.
Such sequences are called geometric sequence or geometric progression abbreviated as `G.P`
`\color{fuchsia} \(★ mathbf\ul"Geometric Progression (G . P.)")`
A sequence `color(blue)(a_1, a_2, a_3, …, a_n, … )` is called geometric progression,
if each term is non-zero and `color(red)(a_(k+1)/a_k = r)` (constant ratio ), for `color(red)(k ≥ 1.)`
`\color{red} ✍️` By letting `color(blue)(a_1 = a)`, we obtain a geometric progression, `color(red)(a, ar, ar^2, ar^3,…...,)`
where `a` is called the `color(blue)"first term"` and `r` is called the `color(blue)"common ratio"` of the G.P.
We shall use the following notations with these formulae:
`color(red)(a) = ` the first term, ` \ \ color(red)(r) = `the common ratio, ` \ \ color(red)(l) =` the last term,
`color(red)(n) =` the numbers of terms,
`color(red)(S_n) =` the sum of n terms.
`color(red)("Let us consider the following sequences:")`
`(i) color(blue)(2,4,8,16,.............,) (ii) color(blue)(1/9 , -1/ 27, 1/81 , -1/243 ..............) (iii) color(blue)(.01,.0001,.000001,..................)`
In `(i)` we have `a_1=2 , a_2/a_1=2, a_3/a_2=2, a_4/a_3=2` and so on. In `(ii)`, we observe, `a_1=1/9 , a_2/a_1=-1/3, a_3/a_2=-1/3 , a_4/a_3=-1/3` and so on.
Similarly, state how do the terms in `(iii)` progress
In `(i),` this constant ratio is `2;` in `(ii)` it is `-1/3` and in `(iii)` the constant ratio is `0.01.`
Common ratio in geometric progression `(i), (ii)` and `(iii)` above are `2, -1/3` and `0.01,` respectively.
Such sequences are called geometric sequence or geometric progression abbreviated as `G.P`