Question

How is the infinite series from n=0 to infinity of 3(2^(n+1))/5^n=10?

I keep trying to use the geometric sum formula but I keep getting 2.

Answer 1

`sum_(n=0)^oo (3*2^(n+1))/5^n = sum_(n=0)^oo (3*2*2^n)/5^n =6 sum_(n=0)^oo 2^n/5^n`

we can now use the geometric series:

`sum_(n=0)^oo 2^n/5^n = sum_(n=0)^oo (2/5)^n`

and as `abs(2/5) <1`:

`sum_(n=0)^oo 2^n/5^n = 1/(1-2/5) = 5/3`

Then:

`sum_(n=0)^oo (3*2^(n+1))/5^n = 6 *5/3 = 10`