Dsider the arithmetic sequence: \( -\frac{7}{2} ;-3 ;-\frac{5}{2} \ldots \) 2 Determine the general term of the sequence. \( \begin{array}{l}\text { The sum of the first } n \text { terms of this sequence is } 675 \text {. } \\ \text { Calculate the value of } n \text {. } \\ \text { A new sequence is formed by squaring each term of the given arithmetic } \\ \text { sequence. Determine which term of the new sequence will bave the } \\ \text { smallest value. }\end{array} \begin{array}{l}\text { (4) }\end{array} \)

Let's analyze the given arithmetic sequence step by step to address each of your questions. **Given Arithmetic Sequence:** \[ -\frac{7}{2},\ -3,\ -\frac{5}{2},\ \ldots \] ### 1. **Determine the General Term of the Sequence** An arithmetic sequence has a general term formula: \[ a_n = a_1 + (n - 1)d \] where: - \( a_n \) = the \( n \)-th term - \( a_1 \) = the first term - \( d \) = common difference - \( n \) = term number **Identifying the Parameters:** - First term (\( a_1 \)) = \( -\frac{7}{2} \) - Second term = \( -3 \) **Calculating the Common Difference (\( d \)):** \[ d = \text{Second term} - \text{First term} = -3 - \left(-\frac{7}{2}\right) = -3 + \frac{7}{2} = -\frac{6}{2} + \frac{7}{2} = \frac{1}{2} \] **General Term Formula:** \[ a_n = -\frac{7}{2} + (n - 1) \cdot \frac{1}{2} = \frac{n - 8}{2} \] So, the general term is: \[ a_n = \frac{n - 8}{2} \] --- ### 2. **Calculate the Value of \( n \) Given \( S_n = 675 \)** **Sum of the First \( n \) Terms of an Arithmetic Sequence:** \[ S_n = \frac{n}{2} (a_1 + a_n) \] Given \( S_n = 675 \), substitute the known values: \[ 675 = \frac{n}{2} \left(-\frac{7}{2} + \frac{n - 8}{2}\right) \] Simplify the expression inside the parentheses: \[ -\frac{7}{2} + \frac{n - 8}{2} = \frac{n - 15}{2} \] Now, plug this back into the sum formula: \[ 675 = \frac{n}{2} \cdot \frac{n - 15}{2} = \frac{n(n - 15)}{4} \] Multiply both sides by 4 to eliminate the denominator: \[ 2700 = n(n - 15) \] Rearrange the equation: \[ n^2 - 15n - 2700 = 0 \] **Solving the Quadratic Equation:** \[ n = \frac{15 \pm \sqrt{(15)^2 + 4 \cdot 2700}}{2} = \frac{15 \pm \sqrt{225 + 10800}}{2} = \frac{15 \pm \sqrt{11025}}{2} = \frac{15 \pm 105}{2} \] This yields two possible solutions: \[ n = \frac{15 + 105}{2} = 60 \quad \text{or} \quad n = \frac{15 - 105}{2} = -45 \] Since \( n \) must be a positive integer, we discard \( -45 \): \[ n = 60 \] **Therefore, the value of \( n \) is 60.** --- ### 3. **Determine Which Term of the New Sequence (Squared Terms) Has the Smallest Value** **Forming the New Sequence by Squaring Each Term:** \[ b_n = (a_n)^2 = \left(\frac{n - 8}{2}\right)^2 = \frac{(n - 8)^2}{4} \] **Analyzing the Squared Sequence:** The expression \( \frac{(n - 8)^2}{4} \) is a quadratic function in terms of \( n \), which represents a parabola opening upwards. The minimum value of this parabola occurs at its vertex. **Finding the Vertex:** For a quadratic function \( f(n) = \frac{(n - 8)^2}{4} \), the vertex occurs at \( n = 8 \). **Conclusion:** - The smallest value of the squared sequence \( b_n \) occurs at \( n = 8 \). - Therefore, the **8th term** of the new sequence has the smallest value. --- **Summary:** 1. **General Term:** \( a_n = \frac{n - 8}{2} \) 2. **Value of \( n \) for \( S_n = 675 \):** \( n = 60 \) 3. **Term with Smallest Value in Squared Sequence:** 8th term