The \(3^{\text {rd }}, 14^{\text {th }}\) and \(69^{\text {th }}\) terms of an arithmetic progression form three distinct and consecutive terms of a geometric progression. If the next term of the…

The next index is $344$. If the arithmetic progression has common difference $d$, the consecutive geometric differences are $11d$ and $55d$, so the geometric ratio is $5$. The next gap is $275d$, giving $69+275=344$, while $275$ gives the increment alone and omits the starting index $69$.