Question

Please help right away.

Answer 1

`\sum\limits_(k=1)^(\infty)420\left((1)/(6)\right)^(k-1)`

The infinite geometric series is converges if |r| < 1.

We have r=1/6 < 1, therefore our infinite geometric series is converges.

The sum S of an infinite geometric series with |r| < 1 is given by the formula :

`S=(a_1)/(1-r)`

We have:

`a_1=420\left((1)/(6)\right)^(1-1)=420\left((1)/(6)\right)^0=420\\\\r=(1)/(6)`

substitute:

`S=(420)/(1-(1)/(6))=(420)/((5)/(6))=420\cdot(6)/(5)=84\cdot6=504`

Answer: d. Converges, 504.