To solve this problem, we need to determine the type of sequence given, analyze its terms, and find a general formula or other missing terms if necessary.

Problem Analysis

The problem gives:

• $a_3 = \frac{2}{5}$
• $a_7 = \frac{6}{5}$

This sequence appears to follow either an arithmetic or geometric progression. Let's determine which one it is by testing both approaches.

1. Test for Arithmetic Sequence

In an arithmetic sequence, each term differs from the previous one by a constant difference $d$. The $n$-th term of an arithmetic sequence is given by: $a_n = a_1 + (n - 1) \cdot d$

Let's denote:

• $a_3 = a_1 + 2d = \frac{2}{5}$
• $a_7 = a_1 + 6d = \frac{6}{5}$

We now have a system of equations:

1. $a_1 + 2d = \frac{2}{5}$
2. $a_1 + 6d = \frac{6}{5}$

Solving the System

Subtract equation 1 from equation 2 to solve for $d$: $(a_1 + 6d) - (a_1 + 2d) = \frac{6}{5} - \frac{2}{5}$ $4d = \frac{4}{5}$ $d = \frac{1}{5}$

Now, substitute $d = \frac{1}{5}$ back into equation 1 to solve for $a_1$: $a_1 + 2 \cdot \frac{1}{5} = \frac{2}{5}$ $a_1 + \frac{2}{5} = \frac{2}{5}$ $a_1 = 0$

Conclusion for the Sequence

Thus, the sequence is arithmetic with:

• First term $a_1 = 0$
• Common difference $d = \frac{1}{5}$

The general term $a_n$ for this sequence can be expressed as: $a_n = 0 + (n - 1) \cdot \frac{1}{5} = \frac{n - 1}{5}$

Final Answer

The general term of the sequence is: $a_n = \frac{n - 1}{5}$

Would you like further details, or do you have any questions?

Here are 5 related questions for further exploration:

1. How would you find the 10th term of this sequence?
2. If given a geometric sequence, how would the approach change?
3. What is the sum of the first 10 terms of this sequence?
4. Can you derive the formula for the sum of an arithmetic sequence?
5. How would you find the common difference if given non-consecutive terms?

Tip: When dealing with sequences, determining if it's arithmetic or geometric is often the first helpful step.