The given sequence is:

$5, 11, 17, 23, \dots$

1. Type of Sequence:

This is an Arithmetic Progression (AP) because the difference between consecutive terms is constant.

2. Common Difference (d):

The common difference $d$ can be calculated as: $d = 11 - 5 = 6 \quad \text{or} \quad d = 17 - 11 = 6$ Thus, the common difference is $d = 6$.

3. General Term (A$_n$):

The general term of an arithmetic sequence is given by: $A_n = A_1 + (n-1) \cdot d$ Where:

• $A_1 = 5$ (the first term),
• $d = 6$ (common difference).

So, $A_n = 5 + (n-1) \cdot 6 = 5 + 6n - 6 = 6n - 1$

4. To find A$_9$ (the 9th term):

$A_9 = 6 \cdot 9 - 1 = 54 - 1 = 53$

5. To find S$_9$ (the sum of the first 9 terms):

The sum of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \cdot (A_1 + A_n)$ For $n = 9$: $S_9 = \frac{9}{2} \cdot (5 + 53) = \frac{9}{2} \cdot 58 = 9 \cdot 29 = 261$

Final Answers:

• Type of sequence: Arithmetic Progression (AP).
• A$_9$: 53.
• S$_9$: 261.

Would you like further details or have any questions?

Related Questions:

1. How would you find the 15th term in this arithmetic sequence?
2. What is the sum of the first 20 terms of this sequence?
3. Can you derive the formula for the $n$-th term in a geometric sequence?
4. How does an arithmetic progression differ from a geometric progression?
5. What is the formula for the sum of an infinite geometric series?

Tip:

When solving arithmetic progression problems, always double-check the common difference and the number of terms to avoid simple mistakes.