https://file.aiquickdraw.com/mathbot/file/1727093472007619vcw17.jpg

The image shows a problem related to an arithmetic sequence. The text in the image (translated from Indonesian) reads:

"Given that the 3rd term and the 16th term of an arithmetic sequence are 13 and 78, respectively. Determine the formula for the \(n\)-th term."

This is a standard arithmetic progression (AP) problem. Let's break it down:

### Known:
- The 3rd term, \( U_3 = 13 \)
- The 16th term, \( U_{16} = 78 \)

### General formula for the \(n\)-th term of an arithmetic sequence:
\[
U_n = a + (n-1) \cdot d
\]
where:
- \(a\) is the first term,
- \(d\) is the common difference,
- \(U_n\) is the \(n\)-th term.

### Using the information:
1. For \( U_3 = 13 \):
\[
a + 2d = 13
\]
2. For \( U_{16} = 78 \):
\[
a + 15d = 78
\]

Now, solving these two equations to find \(a\) and \(d\).

Let me compute the values for you.The solution gives:
- The first term \(a = 3\),
- The common difference \(d = 5\).

### Formula for the \(n\)-th term of the arithmetic sequence:
\[
U_n = 3 + (n-1) \cdot 5
\]
Simplifying:
\[
U_n = 5n - 2
\]

Thus, the formula for the \(n\)-th term is:
\[
U_n = 5n - 2
\]

Would you like further details or have any other questions? 

Here are five related questions for further practice:
1. How do you find the sum of the first \(n\) terms in an arithmetic sequence?
2. If the 5th term of the sequence is 23, what is the common difference?
3. How does an arithmetic sequence differ from a geometric sequence?
4. Can you derive the formula for the sum of an arithmetic sequence?
5. How would the formula change if the first term was 7 and the common difference was 3?

**Tip:** When solving sequences, always try to derive two independent equations from the known terms for easier solving.

Diketahui suku ke-3 dan suku ke-16 barisan aritmatika berturut-turut adalah 13 dan 78. Tentukan rumus suku ke-n.

This problem demonstrates how to find the nth term of an arithmetic sequence when given two specific terms, the 3rd term being 13 and the 16th term being 78. Suitable for Grades 9-10, covering arithmetic sequence concepts and formula derivation.

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