Math Problem Statement

Complete the nth term rule for the quadratic sequence where the positions and corresponding terms are given.

Solution

The sequence provided is quadratic, meaning its nnth term can be expressed in the form:

n2+cnn^2 + cn

where cc is a constant to be determined.

From the table:

Position nn12345
Term410182840

The general formula starts with n2n^2, so we first calculate n2n^2 for each value of nn:

  • For n=1n = 1: n2=1n^2 = 1
  • For n=2n = 2: n2=4n^2 = 4
  • For n=3n = 3: n2=9n^2 = 9
  • For n=4n = 4: n2=16n^2 = 16
  • For n=5n = 5: n2=25n^2 = 25

Next, subtract n2n^2 from each corresponding term to find the remainder:

  • For n=1n = 1: 41=34 - 1 = 3
  • For n=2n = 2: 104=610 - 4 = 6
  • For n=3n = 3: 189=918 - 9 = 9
  • For n=4n = 4: 2816=1228 - 16 = 12
  • For n=5n = 5: 4025=1540 - 25 = 15

This shows that the difference between the term and n2n^2 is increasing by 3 each time, which suggests a linear relationship. The additional term is 3n3n.

Thus, the nnth term rule for this sequence is:

n2+3nn^2 + 3n

Would you like any further clarification?

Here are 5 related questions:

  1. What is the next term in this quadratic sequence?
  2. How do you find the difference between terms in a quadratic sequence?
  3. What role does the constant cc play in the general formula of a quadratic sequence?
  4. Can you derive a general method for finding the quadratic sequence rule from any given set of terms?
  5. How would this method apply to sequences that are not quadratic but cubic?

Tip: When analyzing sequences, always check if differences between terms are constant or if they follow a pattern that hints at a higher-degree polynomial relationship.

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Math Problem Analysis

Mathematical Concepts

Quadratic Sequences
Sequences and Series
Algebra

Formulas

nth term formula: n^2 + cn
Difference method for quadratic sequences

Theorems

Quadratic sequence properties

Suitable Grade Level

Grades 8-10