The fourth term of a geometric progression is 4 and the seventh term is 32/125. Find the first term, the common ratio and the sum of the first nth terms. Hence, deduce the sum to infinity.

common ratio is 2/5

First term is  125/2

The sum of the first n terms is:

The infinite sum is:

Step-by-step explanation:

The 4th term is 4, and the seventh term is 32/125 so we use the definition of nth term of a geometric sequence to create the following two equations:

Now we divide a7 by a4 to get rid of a1 and work on determining the common ratio of the sequence:

So, the common ratio is 2/5

we can now determine the first term:

The sum of the first n terms is given by the formula:

and therefore, the infinite sum is:

2^y = ln 5, or   y(ln 2) = ln 5.   solving for y,   y = (ln 5) / (ln 2).   y=5

step-by-step explanation: