(a) Given sequence is \(\displaystyle{2},{6},{18},{54},{162}\)

Ratio of two consecutive terms is constant. Therefore, sequence is geometric and common ratio is \(\displaystyle{r}=\frac{{6}}{{2}}\), that is \(\displaystyle{r}={3}\)

Next term can be obtained by multiplying the previous term by r. Therefore, next three terms are

\(\displaystyle{162}\times{3}={486}\)

\(\displaystyle{486}\times{3}={1458}\)

\(\displaystyle{1458}\times{3}={4374}\)

(b) Given sequence is \(\displaystyle{1},{8},{15},{22},{29}\)

Difference of two consecutive terms is constant. Therefore, sequence is arithmetic and common difference is \(\displaystyle{d}={8}-{1}={7}\),

Next term can be obtained by adding the previous term by d. Therefore, next three terms are:

\(\displaystyle{29}+{7}={36}\)

\(\displaystyle{36}+{7}={43}\)

\(\displaystyle{43}+{7}={50}\)

(c) Given sequence is \(\displaystyle{11},{15},{19},{23},{27}\)

Difference of two consecutive terms is constant. thus, sequence is arithmetic and common difference is \(d=15-11=4.\)

Next term can be obtained by adding the previous term by d. Therefore, next three terms are:

\(\displaystyle{27}+{4}={31}\)

\(\displaystyle{31}+{4}={35}\)

\(\displaystyle{35}+{4}={39}\)