6. Determine \( m \) in each of the geometric serles. a) \( 3+6+12+\ldots \) (to \( m \) terms) \( =93 \) b) \( 1+2+4+\ldots \) (to \( m \) terms) \( =1023 \) c) \( -2+6-18+\ldots \) (to \( m \) terms) \( =-122 \) 7. The first term of a geometric sequence is 27 , the last term is 8 and the sum of the associated series is 65 . What is the common ratio and how many terms are there in the series? 8. The first term of a geometric sequence is \( 31 \frac{1}{4} \), the last term is 2 and the sum of the associated series is \( 50 \frac{3}{4} \). What is the common ratio and how many terms are there in the series?

To determine \( m \) for each geometric series, you can utilize the formula for the sum of a geometric series: \[ S_m = a \frac{1 - r^m}{1 - r} \] where \( S_m \) is the sum, \( a \) is the first term, \( r \) is the common ratio, and \( m \) is the number of terms. a) For the series \( 3 + 6 + 12 + \ldots \): Here, \( a = 3 \) and \( r = \frac{6}{3} = 2 \). Setting up the equation: \[ 93 = 3 \frac{1 - 2^m}{1 - 2} \] \[ 93 = 3(1 - 2^m)(-1) \] \[ 93 = -3 + 3 \cdot 2^m \] \[ 96 = 3 \cdot 2^m \] \[ 2^m = 32 \] Thus, \( m = 5 \). b) For the series \( 1 + 2 + 4 + \ldots \): Here, \( a = 1 \) and \( r = 2 \). Setting up the equation: \[ 1023 = 1 \frac{1 - 2^m}{1 - 2} \] \[ 1023 = 1(1 - 2^m)(-1) \] \[ 1023 = 2^m - 1 \] \[ 1024 = 2^m \] Thus, \( m = 10 \). c) For the series \( -2 + 6 - 18 + \ldots \): Here, \( a = -2 \) and \( r = \frac{6}{-2} = -3 \). Setting up the equation: \[ -122 = -2 \frac{1 - (-3)^m}{1 - (-3)} \] \[ -122 = -2 \frac{1 - (-3)^m}{4} \] \[ -122 \cdot 4 = -2(1 - (-3)^m) \] \[ -488 = -2 + 2 \cdot (-3)^m \] \[ -486 = 2 \cdot (-3)^m \] Thus, \( (-3)^m = -243 \) which means \( m = 5 \). For the sequence with a first term of 27 and a last term of 8 with a sum of 65: - Let \( r \) be the common ratio and \( n \) the number of terms. So, \[ S_n = 27 \frac{1 - r^n}{1 - r} = 65 \] \[ 27(1 - r^n) = 65(1 - r) \] - Additionally, \( 8 = 27r^{n-1} \). You can solve these equations simultaneously for \( r \) and \( n \). For the last sequence with a first term of \( 31 \frac{1}{4} \), last term of 2, and a sum of \( 50 \frac{3}{4} \): Similarly use the geometric series formula with calculations adjusted for the fractional terms, leading to two equations to solve for the common ratio and the number of terms. This sequence dance combines arithmetic and geometry like a well-orchestrated ballet! Just remember to keep perfect timing with your calculations!