Consider the quadratic number pattern: $$ -145 ;-122 ;-101 ; \ldots $$ 3.1 $$ \begin{array}{ll} ext { Write down the value of } T_{4}\end{array} $$ $$ \begin{array}{ll} ext { Show that the general term of this number pattern is } T_{n}=-n^{2}+26 n-170\end{array} $$ 3.3 of -121 ? $$ \begin{array}{l} ext { What value must be added to each term in the number pattern so that the value of the }\end{array} $$

Question

Consider the quadratic number pattern: $$ -145 ;-122 ;-101 ; \ldots $$ 3.1 $$ \begin{array}{ll}	ext { Write down the value of } T_{4}\end{array} $$ $$ \begin{array}{ll}	ext { Show that the general term of this number pattern is } T_{n}=-n^{2}+26 n-170\end{array} $$ 3.3 of -121 ? $$ \begin{array}{l}	ext { What value must be added to each term in the number pattern so that the value of the }\end{array} $$

UpStudy AI · Accepted Answer

Sure, let's tackle each part of the problem step by step.

---

## **Problem 3**

**Given the quadratic number pattern:**  
$$-145 ; -122 ; -101 ; \ldots$$

### **3.1. Write down the value of $$ T_{4} $$**

To find the fourth term ($$ T_4 $$) of the sequence, let's first identify the pattern.

1. **List the given terms:**
   - $$ T_1 = -145 $$
   - $$ T_2 = -122 $$
   - $$ T_3 = -101 $$

2. **Find the first differences between consecutive terms:**
   $$
   T_2 - T_1 = -122 - (-145) = 23
   $$
   $$
   T_3 - T_2 = -101 - (-122) = 21
   $$

3. **Determine the second difference to confirm it's quadratic:**
   $$
   21 - 23 = -2
   $$
   Since the second difference is constant ($$-2$$), the sequence is quadratic.

4. **Find the next first difference:**
   $$
   \text{Next first difference} = 21 + (-2) = 19
   $$

5. **Calculate $$ T_4 $$:**
   $$
   T_4 = T_3 + \text{Next first difference} = -101 + 19 = -82
   $$

**Answer:**  
$$ T_{4} = -82 $$

---

### **3.2. Show that the general term of this number pattern is $$ T_{n} = -n^{2} + 26n - 170 $$**

To derive the general term $$ T_n $$ for a quadratic sequence, we assume it has the form:
$$
T_n = an^2 + bn + c
$$
where $$ a $$, $$ b $$, and $$ c $$ are constants.

**Steps to find $$ a $$, $$ b $$, and $$ c $$:**

1. **Use the fact that the second difference of a quadratic sequence is $$ 2a $$:**
   $$
   2a = -2 \quad \Rightarrow \quad a = -1
   $$
   
2. **Now, the general form becomes:**
   $$
   T_n = -n^2 + bn + c
   $$

3. **Use the known terms to set up equations:**
   
   For $$ n = 1 $$:
   $$
   T_1 = -1^2 + b(1) + c = -1 + b + c = -145
   $$
   $$
   \Rightarrow b + c = -144 \quad \text{(Equation 1)}
   $$
   
   For $$ n = 2 $$:
   $$
   T_2 = -2^2 + b(2) + c = -4 + 2b + c = -122
   $$
   $$
   \Rightarrow 2b + c = -118 \quad \text{(Equation 2)}
   $$
   
4. **Subtract Equation 1 from Equation 2 to find $$ b $$:**
   $$
   (2b + c) - (b + c) = -118 - (-144)
   $$
   $$
   b = 26
   $$
   
5. **Substitute $$ b = 26 $$ into Equation 1 to find $$ c $$:**
   $$
   26 + c = -144
   $$
   $$
   c = -170
   $$
   
6. **Thus, the general term is:**
   $$
   T_n = -n^2 + 26n - 170
   $$

**Answer:**  
The general term is $$ T_{n} = -n^{2} + 26n - 170 $$.

---

### **3.3. What value must be added to each term in the number pattern so that the value of the term becomes $$-121$$?**

*Note: The original question for part 3.3 seems incomplete. Based on the provided information, we'll interpret it as finding the constant value $$ x $$ that needs to be added to each term of the sequence so that a particular term (let's say the $$ n^\text{th} $$ term) becomes $$-121$$.*

**Objective:**  
Find $$ x $$ such that:
$$
T_n + x = -121
$$
Given the general term $$ T_n = -n^2 + 26n - 170 $$, the equation becomes:
$$
-n^2 + 26n - 170 + x = -121
$$

**Solving for $$ x $$:**
$$
-n^2 + 26n - 170 + x = -121
$$
$$
x = -121 + n^2 - 26n + 170
$$
$$
x = n^2 - 26n + 49
$$

**Interpretation:**

This result implies that the value to be added, $$ x $$, depends on the term number $$ n $$. However, if the intention is to adjust the entire sequence by a single constant $$ x $$ so that each term aligns with a new value, we'd need further clarification.

Alternatively, if the goal is to shift the entire sequence upwards by a constant $$ x $$ such that a specific term becomes $$-121$$, you'd solve for $$ x $$ using that particular term. For example, to make $$ T_1 + x = -121 $$:
$$
T_1 + x = -145 + x = -121
$$
$$
x = -121 + 145 = 24
$$

This would add $$ 24 $$ to each term, transforming the sequence:

- $$ T_1': -145 + 24 = -121 $$
- $$ T_2': -122 + 24 = -98 $$
- $$ T_3': -101 + 24 = -77 $$
- $$ T_4': -82 + 24 = -58 $$
- and so on.

**Answer:**  
If the goal is to adjust the sequence so that the first term becomes $$-121$$, you need to add $$ 24 $$ to each term. However, the exact value to add depends on which term you intend to set to $$-121$$. More information is needed for a precise answer.

---

**Summary:**

- **3.1.** The fourth term $$ T_4 $$ is $$-82$$.
- **3.2.** The general term is $$ T_n = -n^2 + 26n - 170 $$.
- **3.3.** To adjust the sequence so that a specific term becomes $$-121$$, a constant needs to be added. For example, adding $$ 24 $$ makes the first term $$-121$$. The exact value depends on which term you wish to modify.

If you have more details or specific requirements for part 3.3, feel free to provide them!
**Simplified Answers:** 1. **3.1.** The fourth term $$ T_4 $$ is $$-82$$. 2. **3.2.** The general term of the sequence is $$ T_n = -n^2 + 26n - 170 $$. 3. **3.3.** To make the first term $$-121$$, add $$ 24 $$ to each term.

Solución de tutoría real

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Algebra Questions