Since these are terms of a geometric sequence, they have a common ratio so:
[tex] \frac{5x+1}{x+2} = \frac{x+11}{5x+1} \\ 25x^2+10x+1=x^2+13x+22 \\24x^2-3x-21=0 \\ 8x^2-x-7=0 \\ (8x+7)(x-1)=0[/tex]
Therefore x can be equal to -7/8 or 1.
The sum of n terms in a geometric sequence is equal to [tex] \frac{a_1(r^n-1)}{r-1} [/tex]
When x = -7/8, the first term would be 9/8 and the common ratio would be -3.
The sum of the first ten terms would be:
[tex]( \frac{9}{8})[(-3)^{10}-1]/(-3-1) \\ =( \frac{9}{8})( 59,048)/-4 \\ =-16,607.25[/tex]
When x = 1, the first term would be 3 and the ratio would be 2.
[tex]3(2^{10}-1)/(2-1) \\ =3(1023)/1 \\ =3069[/tex]
