1 Determine the sum of the series: \( 1.1 \quad 11+8+5+2+\ldots-166 \). \( 1.2 \quad 1+5+9+13+\ldots+113 \) \( 1.3-12-5+2+9+\ldots+121 \)

To determine the sum of each series, we can analyze their patterns. 1.1 The series \( 11, 8, 5, 2, \ldots, -166 \) is an arithmetic series where the first term \( a = 11 \) and the common difference \( d = -3 \). We can find the number of terms \( n \) using the formula for the \( n \)-th term of an arithmetic sequence \( a_n = a + (n-1)d \). Setting \( a_n = -166 \): \[ -166 = 11 + (n-1)(-3) \\ -166 - 11 = (n-1)(-3) \\ -177 = -3(n-1) \\ n-1 = 59 \\ n = 60 \] Now, we can find the sum \( S_n \) of the first \( n \) terms: \[ S_n = \frac{n}{2} (a + a_n) = \frac{60}{2} (11 - 166) = 30 \times (-155) = -4650 \] So, the sum of the series is \(-4650\). 1.2 The series \( 1, 5, 9, 13, \ldots, 113 \) is also an arithmetic series where the first term \( a = 1 \) and the common difference \( d = 4 \). To find the number of terms \( n \), set \( a_n = 113 \): \[ 113 = 1 + (n-1)(4) \\ 112 = 4(n-1) \\ n-1 = 28 \\ n = 29 \] Now, calculate the sum \( S_n \): \[ S_n = \frac{n}{2} (a + a_n) = \frac{29}{2} (1 + 113) = \frac{29}{2} \times 114 = 29 \times 57 = 1653 \] So, the sum of the series is \( 1653 \). 1.3 The series \(-12, -5, 2, 9, \ldots, 121\) also forms an arithmetic sequence where the first term \( a = -12 \) and the common difference \( d = 7 \). To find \( n \), set \( a_n = 121 \): \[ 121 = -12 + (n-1)(7) \\ 133 = 7(n-1) \\ n - 1 = 19 \\ n = 20 \] Now, we find the sum \( S_n \): \[ S_n = \frac{n}{2} (a + a_n) = \frac{20}{2} (-12 + 121) = 10 \times 109 = 1090 \] So, the sum of the series is \( 1090 \).