Math problem please solve

Question

Find the value of x so that x+2,5×+1 will form a geometric sequence. Find the sum of the first 10 terms of the given sequence

Answer 1

(x+2/(5x+1)= r= common ratio for a GP

a1 = 5x+1

a2 = x+2

a3 = (x^2+4x+4)/(5x+1)

a4 = (x^2+4x+4)(x+2)/(25x^2+10x +1)

an = (5x+1)r^(n-1)= (5x+1)((x+2)/(5x+1))^(n-1)

x could be any value except -1/5 as that makes the denominator = 0

such as x = 1, then the sequence is

6, 3, 3/2, 3/4, 3/8, ... with r= 1/2