Frederickreynolds

Let the constant be x:

By the definition of a geometric sequence, then:  (60 + x)/(100 + x)  =  (100 + x)/(150+ x)

Cross multiplying:  (60 + x)(150 + x)  =  (100 + x)(100 + x)

--->   9000 + 210x + x²  =  10000 + 200x + x²

--->   10x  =   1000

--->   x = 100 

The series is based on following pattern : 

$latex \left ( 1 \right )^{2}+1=2$
$latex \left ( 2 \right )^{2}+1=5$
$latex \left ( 5 \right )^{2}+1=26$
$latex \left ( 26 \right )^{2}+1=677$
Therefore, the next number of this series will be 677.