if 4x+3and 7x+6are consecutive terms of a geometric sequence, find the values(s)x.
sequence and series
answer
To find the values of x in the given problem, we need to determine the common ratio of the geometric sequence. A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant called the common ratio.
In this case, we have two consecutive terms of the geometric sequence: 4x + 3 and 7x + 6. The second term is obtained by multiplying the first term by the common ratio:
(7x + 6) = r * (4x + 3)
Now, let's solve for x by equating the two expressions:
7x + 6 = r * (4x + 3)
Distribute r to the terms inside the parentheses:
7x + 6 = 4rx + 3r
Next, we want to isolate the variables (x) on one side of the equation. To do this, we can move all terms involving x to one side and the remaining terms to the other side:
7x - 4rx = 3r - 6
Factor out x on the left side:
x(7 - 4r) = 3r - 6
Finally, divide both sides of the equation by (7 - 4r) to solve for x:
x = (3r - 6) / (7 - 4r)
So, the value of x in terms of the common ratio (r) is (3r - 6) / (7 - 4r).