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 `r_1` and `r_2,` 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 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`
6. 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.
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 `r_1` and `r_2,` 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 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`
6. 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.