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

Question

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

UpStudy AI · Accepted Answer

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
1. **General Term:** $$ a_n = \frac{n - 8}{2} $$ 2. **Value of $$ n $$:** 60 3. **Smallest Value in Squared Sequence:** 8th term

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