Concept: Arithmetic Sequence 1.1 Consider the arithmetic sequence 1.1 .1 Write down the next term of the 1.1 .2 Determine the $$ 21^{ ext {th }} $$ term of the $$ s e $$ 1.1 .3 If the $$ n^{ ext {th }} $$ term is 148 , determine $$ t $$ 1.1 .4 Calculate the smallest value of $$ n f $$

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Concept: Arithmetic Sequence 1.1 Consider the arithmetic sequence 1.1 .1 Write down the next term of the 1.1 .2 Determine the $$ 21^{	ext {th }} $$ term of the $$ s e $$ 1.1 .3 If the $$ n^{	ext {th }} $$ term is 148 , determine $$ t $$ 1.1 .4 Calculate the smallest value of $$ n f $$

UpStudy AI · Accepted Answer

Certainly! Let's delve into the concept of **Arithmetic Sequences** and address each of your sub-questions step by step. Since specific details about the sequence (like the first term or the common difference) aren't provided, I'll explain the general approach to solving each type of problem related to arithmetic sequences. This way, you can apply the methods to your specific sequence once you have all the necessary information.

---

## **Arithmetic Sequence Overview**

An **arithmetic sequence** is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the **common difference**, denoted by $$ d $$.

### **General Form of an Arithmetic Sequence:**
- **First term ($$ a_1 $$)**: The starting point of the sequence.
- **Common difference ($$ d $$)**: The constant added to each term to get the next term.
- **$$ n^{\text{th}} $$ term ($$ a_n $$)**: Given by the formula:
  $$
  a_n = a_1 + (n - 1) \times d
  $$

---

## **1.1 Arithmetic Sequence Problems**

### **1.1.1 Write Down the Next Term of the Sequence**

**Approach:**
To find the next term in an arithmetic sequence, you add the common difference ($$ d $$) to the current (last known) term.

**Formula:**
$$
\text{Next Term} = a_{\text{current}} + d
$$

**Example:**
If the current term ($$ a_n $$) is 20 and the common difference ($$ d $$) is 3:
$$
\text{Next Term} = 20 + 3 = 23
$$

---

### **1.1.2 Determine the $$ 21^{\text{st}} $$ Term of the Sequence**

**Approach:**
Use the $$ n^{\text{th}} $$ term formula to find the 21st term.

**Formula:**
$$
a_{21} = a_1 + (21 - 1) \times d = a_1 + 20d
$$

**Example:**
If the first term ($$ a_1 $$) is 5 and the common difference ($$ d $$) is 2:
$$
a_{21} = 5 + 20 \times 2 = 5 + 40 = 45
$$

---

### **1.1.3 If the $$ n^{\text{th}} $$ Term is 148, Determine $$ t $$**

*Assumption:* It seems there might be a variable $$ t $$ representing either the term number ($$ n $$) or another parameter like the common difference ($$ d $$). I'll address both scenarios.

**Case 1: Determine the Term Number ($$ n $$)**
- Given $$ a_n = 148 $$, solve for $$ n $$.

**Formula:**
$$
148 = a_1 + (n - 1) \times d
$$
$$
n = \frac{148 - a_1}{d} + 1
$$

**Example:**
If $$ a_1 = 8 $$ and $$ d = 5 $$:
$$
148 = 8 + (n - 1) \times 5
$$
$$
140 = (n - 1) \times 5
$$
$$
n - 1 = 28 \Rightarrow n = 29
$$

**Case 2: Determine the Common Difference ($$ d $$)**
- Given $$ a_n = 148 $$, solve for $$ d $$.

**Formula:**
$$
148 = a_1 + (n - 1) \times d
$$
$$
d = \frac{148 - a_1}{n - 1}
$$

**Example:**
If $$ a_1 = 10 $$ and $$ n = 25 $$:
$$
148 = 10 + 24d
$$
$$
138 = 24d
$$
$$
d = \frac{138}{24} = 5.75
$$

*Please clarify the role of $$ t $$ if it's different from the above interpretations.*

---

### **1.1.4 Calculate the Smallest Value of $$ n $$**

*Assumption:* Typically, finding the smallest value of $$ n $$ involves determining the least term number that satisfies a given condition, such as exceeding a certain value.

**Approach:**
Use the $$ n^{\text{th}} $$ term formula and solve for $$ n $$ based on the condition.

**Example Scenario:**
*Find the smallest $$ n $$ such that $$ a_n \geq 100 $$.*

**Given:**
- First term ($$ a_1 $$) = 7
- Common difference ($$ d $$) = 4

**Solution:**
$$
a_n = 7 + (n - 1) \times 4 \geq 100
$$
$$
(n - 1) \times 4 \geq 93
$$
$$
n - 1 \geq \frac{93}{4} = 23.25
$$
$$
n \geq 24.25
$$

Since $$ n $$ must be an integer:
$$
n = 25
$$

**Conclusion:** The smallest value of $$ n $$ is **25**.

*Note:* For your specific problem, replace the numbers with the values provided in your sequence and the condition given.

---

## **Summary**

- **Next Term:** Add the common difference to the current term.
- **$$ n^{\text{th}} $$ Term:** Use $$ a_n = a_1 + (n - 1)d $$.
- **Solving for Variables:** Rearrange the $$ n^{\text{th}} $$ term formula to solve for the desired variable ($$ n $$, $$ d $$, etc.).
- **Smallest $$ n $$:** Set up an inequality based on the condition and solve for $$ n $$, ensuring it's the smallest integer that satisfies the condition.

If you provide specific values for the first term, common difference, or any other parameters, I can offer more tailored solutions to each of your sub-questions!
To solve arithmetic sequence problems, use the formula $$ a_n = a_1 + (n - 1)d $$. For the next term, add the common difference to the current term. To find the 21st term, plug $$ n = 21 $$ into the formula. If the $$ n^{\text{th}} $$ term is 148, solve for $$ n $$ or $$ d $$ using the formula. To find the smallest $$ n $$, set up an inequality based on the given condition and solve for $$ n $$.

Concept: Arithmetic Sequence 1.1 Consider the arithmetic sequence 1.1 .1 Write down the next term of the 1.1 .2 Determine the \( 21^{\text {th }} \) term of the \( s e \) 1.1 .3 If the \( n^{\text {th }} \) term is 148 , determine \( t \) 1.1 .4 Calculate the smallest value of \( n f \)

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