EXERCISE 6.3 For each of the following sequences, determine the general rule and hence the \( 100^{\text {th }} \) term \( \begin{array}{lll}\text { (a) } 1 ; 4 ; 9 ; 16 ; \ldots & \text { (b) } 2 ; 5 ; 10 ; 17 ; \ldots & \text { (c) } 4 ; 7 ; 12 ; 19 ; \ldots \\ \text { (d) } 5 ; 8 ; 13 ; 20 ; \ldots & \text { (e) } 0 ; 3 ; 8 ; 15 ; \ldots & \text { (f) }-1 ; 2 ; 7 ; 14 ; \ldots \\ \text { (g) } 2 ; 8 ; 18 ; 32 ; \ldots & \text { (h) } 3 ; 12 ; 27 ; 48 ; \ldots & \text { (i)* } 4 ; 9 ; 16 ; 25 ; \ldots \\ \text { (j)* } 0 ; 1 ; 4 ; 9 ; \ldots & \text { (k)* } \frac{1}{2} ; 2 ; \frac{9}{2} ; 8 ; \ldots & \text { (l)* } \frac{1}{4} ; 1 ; \frac{9}{4} ; 4 ; \ldots\end{array} \)

To determine the general rule for each sequence and find the \( 100^{\text{th}} \) term, we will analyze each sequence step by step. ### (a) Sequence: \( 1, 4, 9, 16, \ldots \) This sequence consists of perfect squares: - \( 1 = 1^2 \) - \( 4 = 2^2 \) - \( 9 = 3^2 \) - \( 16 = 4^2 \) **General Rule:** \( a_n = n^2 \) **100th Term:** \[ a_{100} = 100^2 = 10000 \] ### (b) Sequence: \( 2, 5, 10, 17, \ldots \) The differences between terms are: - \( 5 - 2 = 3 \) - \( 10 - 5 = 5 \) - \( 17 - 10 = 7 \) The differences \( 3, 5, 7 \) increase by \( 2 \) each time, indicating a quadratic sequence. **General Rule:** \( a_n = n^2 + n \) **100th Term:** \[ a_{100} = 100^2 + 100 = 10000 + 100 = 10100 \] ### (c) Sequence: \( 4, 7, 12, 19, \ldots \) The differences between terms are: - \( 7 - 4 = 3 \) - \( 12 - 7 = 5 \) - \( 19 - 12 = 7 \) The differences \( 3, 5, 7 \) also increase by \( 2 \). **General Rule:** \( a_n = n^2 + 3 \) **100th Term:** \[ a_{100} = 100^2 + 3 = 10000 + 3 = 10003 \] ### (d) Sequence: \( 5, 8, 13, 20, \ldots \) The differences are: - \( 8 - 5 = 3 \) - \( 13 - 8 = 5 \) - \( 20 - 13 = 7 \) Again, the differences \( 3, 5, 7 \) increase by \( 2 \). **General Rule:** \( a_n = n^2 + 4 \) **100th Term:** \[ a_{100} = 100^2 + 4 = 10000 + 4 = 10004 \] ### (e) Sequence: \( 0, 3, 8, 15, \ldots \) The differences are: - \( 3 - 0 = 3 \) - \( 8 - 3 = 5 \) - \( 15 - 8 = 7 \) The differences \( 3, 5, 7 \) increase by \( 2 \). **General Rule:** \( a_n = n^2 - 1 \) **100th Term:** \[ a_{100} = 100^2 - 1 = 10000 - 1 = 9999 \] ### (f) Sequence: \( -1, 2, 7, 14, \ldots \) The differences are: - \( 2 - (-1) = 3 \) - \( 7 - 2 = 5 \) - \( 14 - 7 = 7 \) The differences \( 3, 5, 7 \) increase by \( 2 \). **General Rule:** \( a_n = n^2 - 1 \) **100th Term:** \[ a_{100} = 100^2 - 1 = 10000 - 1 = 9999 \] ### (g) Sequence: \( 2, 8, 18, 32, \ldots \) The differences are: - \( 8 - 2 = 6 \) - \( 18 - 8 = 10 \) - \( 32 - 18 = 14 \) The differences \( 6, 10, 14 \) increase by \( 4 \). **General Rule:** \( a_n = 2n^2 \) **100th Term:** \[ a_{100} = 2 \cdot 100^2 = 2 \cdot 10000 = 20000 \] ### (h) Sequence: \( 3, 12, 27, 48, \ldots \) The differences are: - \( 12 - 3 = 9 \) - \( 27 - 12 = 15 \) - \( 48 - 27 = 21 \) The differences \( 9, 15, 21 \) increase by \( 6 \). **General Rule:** \( a_n = n^3 - n \) **100th Term:** \[ a_{100} = 100^3 - 100 = 1000000 - 100 = 999900 \] ### (i) Sequence: \( 4, 9, 16, 25, \ldots \) This sequence consists of perfect squares: - \( 4 = 2^2 \) - \( 9 = 3^2 \) - \( 16 = 4^2 \) - \( 25 = 5^2 \) **General Rule:** \( a_n = (n + 1)^2 \) **100th Term:** \[ a_{100} = (100 + 1)^2 = 101^2 = 10201 \] ### (j) Sequence: \( 0, 1, 4, 9, \ldots \) This sequence consists of perfect squares: - \( 0 = 0^2 \) - \( 1 = 1^2 \) - \( 4 = 2^2 \) - \( 9 = 3^2 \) **General Rule:** \( a_n = n^2 \) **100th Term:** \[ a_{100} = 100^2 = 10000 \] ### (k) Sequence: \( \frac{1}{2}, 2, \frac{9}{2}, 8, \ldots \) The differences are: - \( 2 - \frac{1}{2} = \frac{3}{2} \) - \( \frac{9}{2} - 2 = \frac{5}{2} \) - \( 8 - \frac{9}{2} = \frac{7}{2} \) The differences \( \frac{3}{2}, \frac{5}{2}, \frac{7}{2} \) increase by \( 1 \). **General Rule:** \( a_n = \frac{n^2 + n}{2} \) **100th Term:**