Question

If x,4x+3 and 7x+3 are geometric sequence. find the value of x
please help me​

Answer 1

x =
`(-6)/(7+√(13))`

r =
`(1+√(13) )/(2)`

Explanation:

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-one number called the common ratio.

For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2.

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Let's take r=ratio of the geometric sequence

in first term, x is multiplied by
`r^(0)`

First term: x
`r^(0)`

Second term: 4x+3 = x
`r^(1)`

Third term: 7x+3 = x
`r^(2)`

From the last 2 equation, we have to obtain r and x values.

Second equation - First equation:

3x = x r (r - 1)

Let's state: x different from 0, so we can simplify x, obtaining:

3 =
`r^(2)` - r

0 =
`r^(2)` - r - 3

r =
`(1+√(13) )/(2)` (only the positive value of r counts)

so, going to second term of the sequence:

4x+3 = x
`r^(1)` => x (4 -
`(1+√(13) )/(2)` ) = -3

x (
`(7+√(13) )/(2)` ) = -3 => x =
`(-6)/(7+√(13))`

hope it helps..