Why is 8:6 equivalent to 4:3 and 16:12

8:6 is equivalent to 4:3 because if you divide the 8:3 by 2 then you would get 4:3 . And if you multiply the 8:6 by 2 then you would get 16:12

Short 301

Long The definition of a "Geometric Sequence" is available, for instance, at http://www.mathsisfun.com/algebra/sequences-sums-geometric.html : "In a Geometric Sequence each term is found by multiplying the previous term by a constant."

In this case, you are given the first term (43), and the third term (2107) and you have one missing term, specifically the second term. If you divide the third term (2107) by the first term (43) then you get 49. And you know that 49 = 7 * 7. So, you reach the conclusion that you can multiply the first term (43) by 7 to get to the missing term and that you can multiply that missing term again by 7 to get to the third term (2107). That satisfies our "Geometric Sequence" requirement that each term is multiplied by a constant (the constant is 7, in this case).

So, if you multiply 43 by 7, you get 301 (the missing second term). And if you multiplied 301 again by 7 then you would get to the third term that was given in the question (2107).