Thank you for the clarification! I see that my previous approach was incorrect. Let's reevaluate the sequence properly.

Given Sequence:

• $a_1 = 1$
• $a_2 = \sqrt{3}$
• $a_3 = \sqrt{7}$
• $a_4 = ?$
• $a_5 = ?$

Step 1: Find the pattern.

We can try to see if there's a consistent pattern by examining the differences between the terms under the square roots. The terms under the square roots are:

• $a_1^2 = 1$
• $a_2^2 = 3$
• $a_3^2 = 7$

Looking at these values:

• $3 - 1 = 2$
• $7 - 3 = 4$

Notice that the differences between the squares of consecutive terms are increasing by 2. Therefore, the next difference should be $6$ (i.e., following the pattern $2, 4, 6, \dots$).

Step 2: Calculate the missing terms.

• $a_4^2 = 7 + 6 = 13$ Thus, $a_4 = \sqrt{13}$.

Next, the difference should increase by 2 again:

• $a_5^2 = 13 + 8 = 21$ Thus, $a_5 = \sqrt{21}$.

Final Sequence:

The complete sequence is: $1, \sqrt{3}, \sqrt{7}, \sqrt{13}, \sqrt{21}$

General Term:

To find the general term, observe the pattern in the squares of the terms:

• $1, 3, 7, 13, 21$

These numbers form a quadratic sequence. Let the general term be $a_n^2 = n^2 + n$. Then: $a_n = \sqrt{n^2 + n}$

Conclusion:

The general term of the sequence is: $a_n = \sqrt{n^2 + n}$ And the complete sequence is: $1, \sqrt{3}, \sqrt{7}, \sqrt{13}, \sqrt{21}$

Would you like more details on the steps or explanations?

Follow-up Questions:

1. How can you verify a quadratic pattern in a sequence?
2. What happens if the differences between terms form a geometric pattern instead of an arithmetic one?
3. How would you determine the next term in a sequence like this without a known formula?
4. Can you apply this method to find the general term for sequences that don't involve square roots?
5. What is the best way to distinguish between quadratic and cubic sequences?

Tip:

Always check the differences between terms, and if they don't follow a simple arithmetic or geometric pattern, explore second or third differences to uncover a possible quadratic or cubic relationship.