(3) How many terms are there insider the geometric sequence \( -320 ; 160 ;-80 ; \ldots ;-\frac{5}{64} \). (1) Calculate \( T_{8} \). (2) Determine the value of \( k \) if \( T_{k}=-\frac{5}{16} \).

To find how many terms are in the geometric sequence \( -320, 160, -80, \ldots, -\frac{5}{64} \), we first identify the common ratio. The common ratio \( r \) can be calculated as follows: \[ r = \frac{160}{-320} = -\frac{1}{2}. \] The general term of a geometric sequence can be expressed as: \[ T_n = a \cdot r^{n-1}, \] where \( a \) is the first term. Here, \( a = -320 \) and \( r = -\frac{1}{2} \). To find the number of terms \( n \) such that \( T_n = -\frac{5}{64} \): \[ -\frac{5}{64} = -320 \left(-\frac{1}{2}\right)^{n-1}. \] Dividing both sides by \(-320\) gives: \[ \frac{5}{64} = 320 \left(\frac{1}{2}\right)^{n-1}. \] Simplifying: \[ \frac{5}{64} = \frac{320}{2^{n-1}}. \] Next, we can convert 320 into a power of 2: \[ 320 = 5 \times 2^6. \] Thus, rewriting the equation yields: \[ \frac{5}{64} = \frac{5 \times 2^6}{2^{n-1}}. \] Cancelling 5 from both sides gives: \[ \frac{1}{64} = \frac{2^6}{2^{n-1}}. \] Recognizing that \( \frac{1}{64} = \frac{1}{2^6} \) leads us to: \[ 2^{-6} = 2^{6 - (n-1)}. \] This simplifies to: \[ -6 = 6 - n + 1 \implies -6 = 7 - n. \] Solving for \( n \): \[ n = 13. \] 1) To calculate \( T_8 \): \[ T_8 = -320 \left(-\frac{1}{2}\right)^{8-1} = -320 \left(-\frac{1}{2}\right)^7 = -320 \cdot -\frac{1}{128} = \frac{320}{128} = 2.5. \] So \( T_8 = 2.5 \). 2) To find \( k \) where \( T_k = -\frac{5}{16} \): \[ -\frac{5}{16} = -320 \left(-\frac{1}{2}\right)^{k-1}. \] Dividing by \(-320\) gives: \[ \frac{5}{16} = 320 \left(\frac{1}{2}\right)^{k-1}. \] This leads to: \[ \frac{5}{16} = \frac{320}{2^{k-1}}. \] Rewriting 320 as \( 5 \times 2^6 \): \[ \frac{5}{16} = \frac{5 \times 2^6}{2^{k-1}}. \] Cancelling 5 yields: \[ \frac{1}{16} = \frac{2^6}{2^{k-1}}. \] Recognizing that \( \frac{1}{16} = 2^{-4} \) gives: \[ -4 = 6 - (k-1) \implies -4 = 7 - k. \] Solving for \( k \): \[ k = 11. \] In summary, there are 13 terms in the sequence, \( T_8 = 2.5 \), and \( k = 11 \).