Question

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

Answer 1

Answer:

x =  $\frac{-6}{7+\sqrt{13}}$

r =  $\frac{1+\sqrt{13} }{2}$

Step-by-step 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 =  $\frac{1+\sqrt{13} }{2}$  (only the positive value of r counts)

so, going to second term of the sequence:

4x+3 = x $r^{1}$ => x (4 - $\frac{1+\sqrt{13} }{2}$ ) = -3

x ($\frac{7+\sqrt{13} }{2}$ ) = -3  =>  x =  $\frac{-6}{7+\sqrt{13}}$

hope it helps..