Answer: \(S_8=12.5\); \(S_\infty=15\); \(n=14\).

The first term is

\(a=3.\)

The common ratio is

\(r=\frac{2.4}{3}=0.8.\)

(a) The sum of the first \(8\) terms is

\(S_8=\frac{a(1-r^8)}{1-r}.\)

So

\(S_8=\frac{3(1-0.8^8)}{1-0.8}=12.48\ldots.\)

Therefore

\(S_8=12.5\)

to 3 significant figures.

(b) Since \(|r|\lt1\), the sum to infinity is

\(S_\infty=\frac{a}{1-r}=\frac3{1-0.8}=15.\)

(c) We need

\(S_n\gt0.95S_\infty.\)

Using \(S_n=15(1-0.8^n)\),

\(15(1-0.8^n)\gt0.95(15).\)

So

\(1-0.8^n\gt0.95.\)

Hence

\(0.8^n\lt0.05.\)

Taking logarithms,

\(n\gt\frac{\log0.05}{\log0.8}=13.42\ldots.\)

Therefore the least integer value is

\(n=14.\)