Math Problem Statement

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

Solution

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

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

  • 41=34 - 1 = 3
  • 94=59 - 4 = 5
  • 179=817 - 9 = 8
  • 2917=1229 - 17 = 12

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

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

  • 53=25 - 3 = 2
  • 85=38 - 5 = 3
  • 128=412 - 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=1712 + 5 = 17.

  • Adding this to the last term: 29+17=4629 + 17 = 46.

  • The next difference should be 17+6=2317 + 6 = 23.

  • Adding this: 46+23=6946 + 23 = 69.

  • The next difference should be 23+7=3023 + 7 = 30.

  • Adding this: 69+30=9969 + 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!

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Math Problem Analysis

Mathematical Concepts

Sequences
Quadratic Sequences
Algebra

Formulas

Quadratic sequence pattern: Constant second differences

Theorems

Quadratic Growth Theorem

Suitable Grade Level

Grades 7-9