What is the general rule if the first structure is 2,the second is 5 ,the third is 10 and the fifth is 17

To determine the general rule for a sequence given the terms:

- First term: 2
- Second term: 5
- Third term: 10
- Fourth term: (not provided)
- Fifth term: 17

We can look for a pattern or a formula that fits these values. Let's analyze the differences between consecutive terms:

1. **Difference between the 2nd and 1st terms:**
   \[
   5 - 2 = 3
   \]
2. **Difference between the 3rd and 2nd terms:**
   \[
   10 - 5 = 5
   \]
3. **Difference between the 4th (not provided, let's assume it) and 3rd terms:**
   \[
   \text{(Fourth term) - 10}
   \]
4. **Difference between the 5th and 4th terms:**
   \[
   17 - \text{(Fourth term)}
   \]

Given the increasing nature of differences, it suggests that the sequence might be quadratic. 

Assume the general form of the sequence is:

\[
a_n = an^2 + bn + c
\]

We have the following system of equations based on the given terms:

1. \( a(1)^2 + b(1) + c = 2 \)
2. \( a(2)^2 + b(2) + c = 5 \)
3. \( a(3)^2 + b(3) + c = 10 \)
4. \( a(5)^2 + b(5) + c = 17 \)

Substituting in the first three points:

1. \( a + b + c = 2 \)
2. \( 4a + 2b + c = 5 \)
3. \( 9a + 3b + c = 10 \)

Now, let's solve this system of equations.

**Step 1:** Subtract the first equation from the second:

\[
(4a + 2b + c) - (a + b + c) = 5 - 2
\]

\[
3a + b = 3 \quad \text{(Equation 4)}
\]

**Step 2:** Subtract the second equation from the third:

\[
(9a + 3b + c) - (4a + 2b + c) = 10 - 5
\]

\[
5a + b = 5 \quad \text{(Equation 5)}
\]

**Step 3:** Subtract Equation 4 from Equation 5:

\[
(5a + b) - (3a + b) = 5 - 3
\]

\[
2a = 2 \quad \Rightarrow \quad a = 1
\]

**Step 4:** Substitute \( a = 1 \) into Equation 4:

\[
3(1) + b = 3 \quad \Rightarrow \quad b = 0
\]

**Step 5:** Substitute \( a = 1 \) and \( b = 0 \) into the first equation:

\[
1(1) + 0(1) + c = 2 \quad \Rightarrow \quad c = 1
\]

Thus, the general formula for the sequence is:

\[
a_n = n^2 + 1
\]

This formula matches all the given terms:

- \( a_1 = 1^2 + 1 = 2 \)
- \( a_2 = 2^2 + 1 = 5 \)
- \( a_3 = 3^2 + 1 = 10 \)
- \( a_4 = 4^2 + 1 = 17 \)

So the **general rule** for the sequence is \( n^2 + 1 \).

Would you like more details or have any questions? Here are some related questions:

1. What is the 10th term in the sequence \( n^2 + 1 \)?
2. How would the sequence change if the first term was different?
3. Can you derive the sum of the first \( n \) terms of this sequence?
4. What is the relationship between this sequence and arithmetic progressions?
5. How does the rate of growth compare between \( n^2 + 1 \) and other polynomial sequences?

**Tip:** For sequences where the differences between consecutive terms are not constant, try looking for quadratic or higher-order polynomials as the general rule.

Discover the general rule for a sequence with terms 2, 5, 10, and 17 through detailed analysis and quadratic function derivation. Explore step-by-step solutions and mathematical concepts suitable for Grades 10-12.

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