The sum of the first and third term of a geometric sequence is 72. The sum to infinity of this sequence is 360, find the possible values of the common ratio, r.
We are dealing with a geometric sequence here, we will therefore focus on the geometric sequence formulas that are given in the math HL formula booklet. The first part of the information we are given states that adding the first and third term of the sequence together is 72. Looking at the formula booklet, there is a formula that allows you to write any term in a geometric series in terms of the first term and common ratio. Instead of writing u1+u3 = 72, we can write u1+ u1r2 = 72. The second expression is much better because it contains variables that we want to solve i.e r and u1.We still do not have enough information to solve this problem as we have 1 equation and 2 unknowns, so we have to figure out a way to find a second equation that will satisfy this problem. Looking at the remaining information we were given, the sum to infinity of this sequence is 360. Looking at the geometric sequence formulas in the formula booklet, we can quickly find an equation that relates the infinite sum of a geometric sequence with more meaningful variables, S = u1/(1-r). Putting the information we know into the equation, we get the following equation: 360 = u1/(1-r). Another equation with only two unknowns! We finally have two equations to solve this problem and it is simple algebra from here!
