Find the sum of \(n\) terms of the geometric progression \[\frac{1}{2}+\frac{1}{6}+\frac{1}{18}+\dotsb.\]

What is the common ratio of this geometric series?

What is the \(n\)th term of this geometric series?

Let \(S_n\) denote the sum of the first \(n\) terms. Could we write the expansion of \(\frac{1}{3}S_n\) underneath the expression for \(S_n\)?

What do we notice? What happens if we take the two expressions away from each other?

Deduce the sum to infinity of this series.

When \(n\to \infty\), what does \(S_n\) tend towards?

Find the least number of terms of the series which must be taken for their sum to exceed \(\dfrac{2999}{4000}\).

Could we try to find an \(n\) such that \(S_n = \dfrac{2999}{4000}\)?

Could we write down the first few powers of \(3\)?