For a geometric sequence, a+ ar+ ar^2+ ar^2+ ..., the 2nd term is ar= 36 and the 4th term is ar^2= 16. Solve those two equations for a and r. You can start by eliminating "a" bydividingone equation by another.

What you give is NOT an infinite series so there is no question of convergence! Do you mean that the series continues a (x-1)^n? That's a "power series" and you can use the "ratio test" to determine for which values of x it converges.2. The floowing series is given :

(x-1) + (x-1)^{2 }+ (x-1)^{3}

2.1 for which values of x will the series converge ?

Again, what you have written makes no sense. Your sum has k going from 1 to n so "where k= 1" is meaningless.3.

n

Σ(3-4k) = -125

k=1

where k =1. determine the value of n.

Dropping the "where k= 1", just go ahead and do the arithetic. If n= 1, the "sum" is just 3- 4= -1, not -125. If n= 2, the sum is (3- 4)+ (3- 8)= -1- 5= -6. Keep going.