Find the value of A for which 1+e^a + e^2a+...= 12 PLEASE HELP?!

This is for calc 2. please show your work im very confused


Find the value of A for which 1+e^a + e^2a+...= 12

Note that 1 + e^a + e^(2a) + ... is a geometric series with a first term of 1 and a common ratio of e^a.

Assuming that e^a < 1 ==> a < 0, we see that:
1 + e^a + e^(2a) + ... = 1/(1 - e^a).

So, we need to solve:
1/(1 - e^a) = 12.

Solving yields:
1/(1 - e^a) = 12
==> 1 = 12(1 - e^a)
==> 1 = 12 - 12e^a
==> 12e^a = 11
==> e^a = 11/12 (note that e^a < 1, so this is a valid solution)
==> a = ln(11/12).

I hope this helps!