10 The sum of the first \( m \) terms of an arithmetic sequence with first term -5 and common difference 4 is 660 . Find \( m \).

To find \( m \), we can use the formula for the sum of the first \( m \) terms of an arithmetic sequence, which is given by: \[ S_m = \frac{m}{2} (2a + (m - 1)d) \] Where \( S_m \) is the sum, \( a \) is the first term, \( d \) is the common difference, and \( m \) is the number of terms. Here, \( S_m = 660 \), \( a = -5 \), and \( d = 4 \). Plugging in these values, we have: \[ 660 = \frac{m}{2} (2(-5) + (m - 1)4) \] This simplifies to: \[ 660 = \frac{m}{2} (-10 + 4m - 4) \] \[ 660 = \frac{m}{2} (4m - 14) \] Multiplying both sides by 2 to eliminate the fraction: \[ 1320 = m(4m - 14) \] Rearranging gives us a quadratic equation: \[ 4m^2 - 14m - 1320 = 0 \] Now, we can simplify this equation by dividing all terms by 2: \[ 2m^2 - 7m - 660 = 0 \] To solve for \( m \), we can use the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -7 \), and \( c = -660 \): First, calculate the discriminant: \[ b^2 - 4ac = (-7)^2 - 4(2)(-660) = 49 + 5280 = 5329 \] Now we find the square root of the discriminant: \[ \sqrt{5329} = 73 \] Applying the quadratic formula: \[ m = \frac{-(-7) \pm 73}{2(2)} = \frac{7 \pm 73}{4} \] Calculating the two potential values for \( m \): 1. \( m = \frac{80}{4} = 20 \) 2. \( m = \frac{-66}{4} = -16.5 \) (not valid since \( m \) must be a positive integer) Therefore, the only valid solution is: \[ \boxed{20} \]