For each of the following number patterns, determine the general term \( \left(n^{\text {th }}\right. \) term) and hence the \( 10^{\text {th }} \) term. \( \begin{array}{lll}\text { (a) } 2 ; 4 ; 8 ; 16 ; \ldots & \text { (b) } 1 ; 3 ; 9 ; 27 ; \ldots & \text { (c) } 4 ; 12 ; 36 ; \ldots \\ \text { (d) } 5 ; 10 ; 20 ; 40 ; \ldots & \text { (e) } 2 ; 8 ; 32 ; \ldots & \text { (f) } 3 ; 9 ; 27 ; \ldots\end{array} \)

**(a) Sequence: \(2,\,4,\,8,\,16,\,\ldots\)** The sequence is geometric with first term \[ a=2 \quad \text{and common ratio} \quad r=2. \] The general term is: \[ a_n = 2 \cdot 2^{n-1} = 2^n. \] The \(10^{\text{th}}\) term is: \[ a_{10} = 2^{10} = 1024. \] **(b) Sequence: \(1,\,3,\,9,\,27,\,\ldots\)** This is a geometric sequence with \[ a=1 \quad \text{and} \quad r=3. \] The general term is: \[ a_n = 1 \cdot 3^{n-1} = 3^{n-1}. \] Thus, the \(10^{\text{th}}\) term is: \[ a_{10} = 3^{9} = 19683. \] **(c) Sequence: \(4,\,12,\,36,\,\ldots\)** This is a geometric sequence with \[ a=4 \quad \text{and} \quad r=3. \] The general term is: \[ a_n = 4 \cdot 3^{n-1}. \] The \(10^{\text{th}}\) term is: \[ a_{10} = 4 \cdot 3^{9} = 4 \times 19683 = 78732. \] **(d) Sequence: \(5,\,10,\,20,\,40,\,\ldots\)** This is a geometric sequence with \[ a=5 \quad \text{and} \quad r=2. \] The general term is: \[ a_n = 5 \cdot 2^{n-1}. \] The \(10^{\text{th}}\) term is: \[ a_{10} = 5 \cdot 2^{9} = 5 \times 512 = 2560. \] **(e) Sequence: \(2,\,8,\,32,\,\ldots\)** This is a geometric sequence with \[ a=2 \quad \text{and} \quad r=4. \] The general term is: \[ a_n = 2 \cdot 4^{n-1}. \] The \(10^{\text{th}}\) term is: \[ a_{10} = 2 \cdot 4^{9}. \] Since \[ 4^{9} = 262144, \] we have \[ a_{10} = 2 \times 262144 = 524288. \] **(f) Sequence: \(3,\,9,\,27,\,\ldots\)** This sequence is geometric with \[ a=3 \quad \text{and} \quad r=3. \] The general term is: \[ a_n = 3 \cdot 3^{n-1} = 3^n. \] Thus, the \(10^{\text{th}}\) term is: \[ a_{10} = 3^{10} = 59049. \]