Find the sixth, seventh, and eighth term of the following sequence:  1,4,9,17,29

The given sequence is: 1, 4, 9, 17, 29.

Let's identify the pattern by examining the differences between consecutive terms:

- \( 4 - 1 = 3 \)
- \( 9 - 4 = 5 \)
- \( 17 - 9 = 8 \)
- \( 29 - 17 = 12 \)

The differences between consecutive terms are: 3, 5, 8, 12.

Now, let's look at the differences between these differences:

- \( 5 - 3 = 2 \)
- \( 8 - 5 = 3 \)
- \( 12 - 8 = 4 \)

The second differences are increasing by 1. This indicates that the sequence follows a quadratic pattern.

### Finding the next terms:
- The next difference should be \( 12 + 5 = 17 \).
- Adding this to the last term: \( 29 + 17 = 46 \).

- The next difference should be \( 17 + 6 = 23 \).
- Adding this: \( 46 + 23 = 69 \).

- The next difference should be \( 23 + 7 = 30 \).
- Adding this: \( 69 + 30 = 99 \).

Thus, the sixth, seventh, and eighth terms are:
- Sixth term = 46
- Seventh term = 69
- Eighth term = 99

Would you like further details or have any questions?

Here are 5 related questions:
1. How can you find the nth term of a quadratic sequence?
2. What are the first and second differences in a sequence?
3. How do you derive a general formula for a quadratic sequence?
4. What are some real-world applications of quadratic sequences?
5. How do linear and quadratic sequences differ in terms of growth?

**Tip:** To identify quadratic sequences, always check for constant second differences!

Discover how to find the sixth, seventh, and eighth terms in the quadratic sequence 1, 4, 9, 17, 29. This step-by-step solution explains the use of first and second differences to identify the quadratic pattern. Ideal for students in Grades 7-9 learning about sequences and quadratic growth.

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