I had a Sequences question that was proving very hard for me

Question

The sum of the first three terms of a geometric progression is 126. If 14, 36, and 4 are added to each of these terms, respectively, then the new numbers form an arithmetic progression. Find the sixth term of the geometric progression.

Answer 1

The sequences are:

geometric- a, ra , r²a 

arithmetic- a+14, ra + 36, r²a + 4

 

The equations are   a + ra + r²a =126

and ra+ 36 - (a + 14) = r²a +4 - (ra + 36)
 I leave you to solve these two equations for a and r.

You should obtain

a = 6, r = 4

The sixth term is ar⁵. Substitute to obtain answer.