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.