Let's find the general term of the sequence $1, \sqrt{3}, \sqrt{7}, \dots$ and determine the missing terms.

Step 1: Identify the pattern.

The sequence begins with:

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

Let's define a general term for the sequence. Assume the general term takes the form $a_n = \sqrt{An + B}$, where $A$ and $B$ are constants we need to determine.

Step 2: Set up equations based on known terms.

From the given sequence, we have:

1. $a_1 = \sqrt{A(1) + B} = 1$
2. $a_2 = \sqrt{A(2) + B} = \sqrt{3}$
3. $a_3 = \sqrt{A(3) + B} = \sqrt{7}$

We can square both sides of each equation to eliminate the square roots:

1. $A(1) + B = 1^2 = 1$
2. $A(2) + B = (\sqrt{3})^2 = 3$
3. $A(3) + B = (\sqrt{7})^2 = 7$

This results in the system of equations:

1. $A + B = 1$
2. $2A + B = 3$
3. $3A + B = 7$

Step 3: Solve the system of equations.

We can subtract the first equation from the second and third equations to eliminate $B$:

• $(2A + B) - (A + B) = 3 - 1$
• $A = 2$

Now subtract the second equation from the third:

• $(3A + B) - (2A + B) = 7 - 3$
• $A = 4$

Using $A = 2$, substitute into the first equation:

• $2 + B = 1$
• $B = -1$

Step 4: General term.

Now that we have $A = 2$ and $B = -1$, the general term is: $a_n = \sqrt{2n - 1}$

Step 5: Find the missing terms.

Now, using the general term formula $a_n = \sqrt{2n - 1}$, let's calculate the next terms:

• $a_4 = \sqrt{2(4) - 1} = \sqrt{8 - 1} = \sqrt{7}$
• $a_5 = \sqrt{2(5) - 1} = \sqrt{10 - 1} = 3$

Final Sequence:

The full sequence is: $1, \sqrt{3}, \sqrt{7}, \sqrt{7}, 3$

Would you like further clarification or a more detailed explanation on any part of this process?

Follow-up Questions:

1. How would you approach solving a different sequence problem with non-square root terms?
2. Can you derive the general term if the sequence starts with different values?
3. How do arithmetic sequences differ from this type of sequence?
4. How would the general term change if the difference between terms was constant?
5. What if the general term involves a cubic or higher-degree polynomial?

Tip:

When analyzing sequences, always check if differences between consecutive terms follow a recognizable pattern, such as arithmetic or geometric, or involve square roots or other functions.