The sequence \left(a_{n}\right) satisfies a_{1}=1 and 5^{\left(a_{n+1}-a_{n}\right)}-1=\frac{1}{n+\frac{2}{3}} for n \geq 1. Let k be the least integer greater than 1 for which a_{k} is an integer. Find k.
The sequence \left(a_{n}\right) satisfies a_{1}=1 and 5^{\left(a_{n+1}-a_{n}\right)}-1=\frac{1}{n+\frac{2}{3}} for n \geq 1. Let k be the least integer greater than 1 for which a_{k} is an integer. Find k.