In a sequence if each term (except the first non zero term) bears the same constant ratio with its immediately preceding term then the series is called a `G.P.` and the constant ratio is called the `text(common ratio.)` Standard appearance of a `GP.` is
`a,ar,ar^2,ar^3,.........,ar^(n-1)`
`text(General term/)``n^t(h)` `text(term/Last term of )``G.P.`
It is given by `T_n = a . r^(n-1)`
where `a=` first term, `r=` common ratio and `n =` position of the term which we required.
`text( Properties of Geometric Progression :)`
1. If `a_1 , a_2 , a_3 ........` are in GP with common ratio r, then `a_1k , a_2k , a_3k,........` and
`a_1/k ,a_2/k ,a_3/k , ..........` are also in GP `(k ne 0)` with common ratio r.
2. If `a_1, a_2 , a_3, ...` are in GP with common ratio r, then
`a_1 pm k , a_2 pm k , a_3 pm k , ...........` are not in GP `(k ne 0).`
3. If `a_1, a_2 , a_3, ...` are in GP with common ratio r, then
`(i) 1/a_1 , 1/a_2 . 1/a_3 ............` are also GP common ratio `1/r`
`(ii) a_1^n , a_2^n , a_3^n ,............` are also in GP with common ratio `r^n` and `n in Q.`
`(iii) loga_1 , loga_2 , loga_3 , .............` are in AP `(a_1 > 0, AA i)`
In this case, the converse also holds good.
4. If `a_1, a_2 , a_3, ...` and `b_1, b_2 , b_3, ...` are two GP's with common ratios r1 and r2,
respectively. Then,
`(i) a_1b_1 , a_2b_2 , a_3b_3 , ..........` and `a_1/b_1 , a_2/b_2 , a_3/b_3,.............` are also in GP with common
ratios `r_1r_2` and `r_1/r_2` respectively.
`(ii)` `a_1 pm b_1 , a_2 pm b_2 , a_3 pm b_3 ,.............` are not in GP.
5. If `a_1 , a_2 , a_3 , ............. , a_(n-2) ,a_(n-1),a_n` are in GP. Then,
`(i) a_1a_n = a_2a_(n-1) = a_3n_(n-2)`
`(ii) a_r = sqrt(a_(r-k) a_(r+k)) , AA k ,0 <= h <= n -r`
`(iii) a_2/a_1 = a_3/a_2 = a_4/a_3 = ......... = a_n/a_(n-1)`
`=> a_2^2 + a_3a_1 , a_3^2 = a_2a_4 ,..........`
Also `a_2 = a_1r , a_3 = a_1 r^2,`
`a_4 = a_1r^3 , .......... , a_n = a_1r^(n-1)`
where, r is the common ratio of GP.
6. If three numbers in GP whose product is given are to be taken as `a/r , a,ar`
and if five numbers in GP whose product is given are to be taken as
`a/r^2,a/r ,a , ar ,ar^2 ,..etc`
`text(In general)` If (2m + 1) numbers in GP whose product is given are to be
taken as `(m in N)`
`a/r^m , a/r^(m-1) , ........ , a/r , a , ar , ............. , ar^(m-1) , ar^m`
7. If four numbers in GP whose product is given are to be taken as
`a/r^3 , a/r , ar , ar^3` and if six numbers in GP whose product is given are to be
taken as `a/r^5 , a/r^3 , a/r , ar , ar^3, ar^5,` etc.
`text(In general)` If(2m) numbers in GP whose product is given are to he taken
as `(m in N)`
`a/r^(2m-1) , a/r^(2m-3) , .......... , a/r^3, a/r , ar , ar^3 , ........... , ar^(2m-3), ar^(2m-1)`
`text(Geometric Mean) (G.M.) :`
If `a, b, c` are three positive numbers in `G.P.` then `b` is called the `text(geometrical mean)` between `a` and `c` and `b^2= ac.` If `a` and `b` are two positive real numbers and `G` is the `G.M.` between them, then
`quadquadquadquadquadquadquadG^2 = ab`
In a sequence if each term (except the first non zero term) bears the same constant ratio with its immediately preceding term then the series is called a `G.P.` and the constant ratio is called the `text(common ratio.)` Standard appearance of a `GP.` is
`a,ar,ar^2,ar^3,.........,ar^(n-1)`
`text(General term/)``n^t(h)` `text(term/Last term of )``G.P.`
It is given by `T_n = a . r^(n-1)`
where `a=` first term, `r=` common ratio and `n =` position of the term which we required.
`text( Properties of Geometric Progression :)`
1. If `a_1 , a_2 , a_3 ........` are in GP with common ratio r, then `a_1k , a_2k , a_3k,........` and
`a_1/k ,a_2/k ,a_3/k , ..........` are also in GP `(k ne 0)` with common ratio r.
2. If `a_1, a_2 , a_3, ...` are in GP with common ratio r, then
`a_1 pm k , a_2 pm k , a_3 pm k , ...........` are not in GP `(k ne 0).`
3. If `a_1, a_2 , a_3, ...` are in GP with common ratio r, then
`(i) 1/a_1 , 1/a_2 . 1/a_3 ............` are also GP common ratio `1/r`
`(ii) a_1^n , a_2^n , a_3^n ,............` are also in GP with common ratio `r^n` and `n in Q.`
`(iii) loga_1 , loga_2 , loga_3 , .............` are in AP `(a_1 > 0, AA i)`
In this case, the converse also holds good.
4. If `a_1, a_2 , a_3, ...` and `b_1, b_2 , b_3, ...` are two GP's with common ratios r1 and r2,
respectively. Then,
`(i) a_1b_1 , a_2b_2 , a_3b_3 , ..........` and `a_1/b_1 , a_2/b_2 , a_3/b_3,.............` are also in GP with common
ratios `r_1r_2` and `r_1/r_2` respectively.
`(ii)` `a_1 pm b_1 , a_2 pm b_2 , a_3 pm b_3 ,.............` are not in GP.
5. If `a_1 , a_2 , a_3 , ............. , a_(n-2) ,a_(n-1),a_n` are in GP. Then,
`(i) a_1a_n = a_2a_(n-1) = a_3n_(n-2)`
`(ii) a_r = sqrt(a_(r-k) a_(r+k)) , AA k ,0 <= h <= n -r`
`(iii) a_2/a_1 = a_3/a_2 = a_4/a_3 = ......... = a_n/a_(n-1)`
`=> a_2^2 + a_3a_1 , a_3^2 = a_2a_4 ,..........`
Also `a_2 = a_1r , a_3 = a_1 r^2,`
`a_4 = a_1r^3 , .......... , a_n = a_1r^(n-1)`
where, r is the common ratio of GP.
6. If three numbers in GP whose product is given are to be taken as `a/r , a,ar`
and if five numbers in GP whose product is given are to be taken as
`a/r^2,a/r ,a , ar ,ar^2 ,..etc`
`text(In general)` If (2m + 1) numbers in GP whose product is given are to be
taken as `(m in N)`
`a/r^m , a/r^(m-1) , ........ , a/r , a , ar , ............. , ar^(m-1) , ar^m`
7. If four numbers in GP whose product is given are to be taken as
`a/r^3 , a/r , ar , ar^3` and if six numbers in GP whose product is given are to be
taken as `a/r^5 , a/r^3 , a/r , ar , ar^3, ar^5,` etc.
`text(In general)` If(2m) numbers in GP whose product is given are to he taken
as `(m in N)`
`a/r^(2m-1) , a/r^(2m-3) , .......... , a/r^3, a/r , ar , ar^3 , ........... , ar^(2m-3), ar^(2m-1)`
`text(Geometric Mean) (G.M.) :`
If `a, b, c` are three positive numbers in `G.P.` then `b` is called the `text(geometrical mean)` between `a` and `c` and `b^2= ac.` If `a` and `b` are two positive real numbers and `G` is the `G.M.` between them, then
`quadquadquadquadquadquadquadG^2 = ab`