# n-th Terms of Quadratic Sequences

## n-th Terms of Quadratic Sequences

• A quadratic sequence is a sequence in which the second differences between terms are constant. This signifies that the terms follow a pattern based on a quadratic formula.

• To find the n-th term of a quadratic sequence, you will need to express the sequence in the form `an^2 + bn + c`.

# Determining the Coefficients of the Quadratic Sequence

• The coefficient `a` in the n-th term is usually half of the second difference in the sequence.

• To find the coefficient `b`, subtract `an^2` from each term in the original sequence consequently forming a linear sequence. The common difference in this new sequence will be `b`.

• To find the coefficient `c`, substitute `n = 1` into the expression `an^2 + bn + c` and set it equal to the first term of the sequence. Subsequently, solve for `c`.

# Application

• After finding `a`, `b`, and `c`, you have the n-th term rule for the quadratic sequence. It allows you to find any term in the sequence without listing all the preceding terms.

• Example: If a sequence has a second difference of 4 and starts with 2, 6, 12, the n-th term rule is `2n^2 - n`.

# Considerations

• Always double-check your coefficients, as even small mistakes can lead to incorrect results.
• Other sequences such as cubic sequences or Fibonacci sequences require different approaches.