2 The eighth and tenth terms of a sequence are 64 and 100. If the sequence is either arithmetic or geometric, the ninth term can not be \( \begin{array}{ll}\text { 1) }-82 & \text { 3) } 80 \\ \text { 2) }-80 & \text { 4) } 82\end{array} \)

To determine which of the given options cannot be the ninth term of the sequence, we will analyze both the arithmetic and geometric cases. ### Case 1: Arithmetic Sequence In an arithmetic sequence, the \(n\)-th term can be expressed as: \[ a_n = a + (n-1)d \] where \(a\) is the first term and \(d\) is the common difference. Given: - \(a_8 = a + 7d = 64\) - \(a_{10} = a + 9d = 100\) We can set up the following equations: 1. \(a + 7d = 64\) (1) 2. \(a + 9d = 100\) (2) Subtract equation (1) from equation (2): \[ (a + 9d) - (a + 7d) = 100 - 64 \] \[ 2d = 36 \implies d = 18 \] Now substitute \(d\) back into equation (1): \[ a + 7(18) = 64 \] \[ a + 126 = 64 \implies a = 64 - 126 = -62 \] Now we can find the ninth term \(a_9\): \[ a_9 = a + 8d = -62 + 8(18) = -62 + 144 = 82 \] ### Case 2: Geometric Sequence In a geometric sequence, the \(n\)-th term can be expressed as: \[ a_n = ar^{n-1} \] where \(a\) is the first term and \(r\) is the common ratio. Given: - \(a_8 = ar^7 = 64\) - \(a_{10} = ar^9 = 100\) We can set up the following equations: 1. \(ar^7 = 64\) (1) 2. \(ar^9 = 100\) (2) Dividing equation (2) by equation (1): \[ \frac{ar^9}{ar^7} = \frac{100}{64} \] \[ r^2 = \frac{100}{64} = \frac{25}{16} \implies r = \frac{5}{4} \text{ or } r = -\frac{5}{4} \] Now substitute \(r\) back into equation (1) to find \(a\): \[ ar^7 = 64 \] For \(r = \frac{5}{4}\): \[ a\left(\frac{5}{4}\right)^7 = 64 \implies a \cdot \frac{78125}{16384} = 64 \implies a = 64 \cdot \frac{16384}{78125} = \frac{1048576}{78125} \] For \(r = -\frac{5}{4}\): \[ a\left(-\frac{5}{4}\right)^7 = 64 \implies a \cdot -\frac{78125}{16384} = 64 \implies a = -64 \cdot \frac{16384}{78125} = -\frac{1048576}{78125} \] Now we can find the ninth term \(a_9\): \[ a_9 = ar^8 \] For \(r = \frac{5}{4}\): \[ a_9 = a \left(\frac{5}{4}\right)^8 = \frac{1048576}{78125} \cdot \frac{390625}{65536} = \frac{40960000000}{5120000000} = 80 \] For \(r = -\frac{5}{4}\): \[ a_9 = a \left(-\frac{5}{4}\right)^8 = -\frac{1048576}{78125} \cdot \frac{390625}{65536} = -80 \] ### Summary of Possible Ninth Terms - From the arithmetic sequence, \(a_9 = 82\). - From the geometric sequence, \(a_9\) can be \(80\) or \(-80\). ### Conclusion The ninth term can be \(80\) or \(-80\) in the geometric case, and \(82\) in the arithmetic case. The only option that cannot be a ninth term is: \[ \boxed{-82} \]