Find the indicated number of missing terms (means) in each given sequence: \( (5 \) points each) a. \( \quad 5 \) arithmetic means between \( \log _{7}(144) \) and \( \log _{7}\left(\frac{9}{4}\right) \)

To find the 5 arithmetic means between \( \log_{7}(144) \) and \( \log_{7}\left(\frac{9}{4}\right) \), we first need to determine the distance between the two given logarithmic values. We can express the sequence of 7 terms as: - \( a_0 = \log_{7}(144) \) - \( a_1 \) (the 1st mean) - \( a_2 \) (the 2nd mean) - \( a_3 \) (the 3rd mean) - \( a_4 \) (the 4th mean) - \( a_5 \) (the 5th mean) - \( a_6 = \log_{7}\left(\frac{9}{4}\right) \) The common difference \( d \) of the arithmetic sequence can be calculated as: \[ d = \frac{a_6 - a_0}{6} \] Calculating \( a_0 \) and \( a_6 \): - \( a_0 = \log_{7}(144) = \log_{7}(12^2) = 2\log_{7}(12) \) - \( a_6 = \log_{7}\left(\frac{9}{4}\right) = \log_{7}(9) - \log_{7}(4) = 2\log_{7}(3) - 2\log_{7}(2) = 2(\log_{7}(3) - \log_{7}(2)) \) Now we substitute these into the formula for \( d \): \[ d = \frac{2(\log_{7}(3) - \log_{7}(2)) - 2\log_{7}(12)}{6} \] \[ d = \frac{2(\log_{7}(3) - \log_{7}(2) - \log_{7}(12))}{6} = \frac{1}{3}(\log_{7}(3) - \log_{7}(2) - \log_{7}(12)) \] Next, we can determine each of the arithmetic means: - \( a_1 = a_0 + d \) - \( a_2 = a_0 + 2d \) - \( a_3 = a_0 + 3d \) - \( a_4 = a_0 + 4d \) - \( a_5 = a_0 + 5d \) This provides all 5 arithmetic means between the given logarithmic values. The numerical evaluations can be performed to find each of the means accordingly.