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A quadratic function is a mathematical function that can be represented in the standard form as \( f(x) = ax^2 + bx + c \). Here, the highest power of the variable \( x \) is 2, which gives the function its characteristic parabolic shape when graphed.
The function can open upwards or downwards depending on the coefficient \( a \). If \( a > 0 \), the parabola opens upwards, while a negative \( a \) opens it downwards.
In the context of sequences like the one given, where \( b_n = n^2 + 4 \), the quadratic nature means each term is generated not just by a simple calculation involving addition or multiplication, but by a square computation. This square term implies a non-linear growth in the terms of the sequence as \( n \) increases.

Sequence evaluation involves substituting different values into a given function to determine the terms of a sequence. This process helps in understanding the pattern or rule that defines the sequence.
For example, consider the sequence defined by \( b_n = n^2 + 4 \).

• At \( n = 0 \), the term is \( b_0 = 0^2 + 4 = 4 \).
• At \( n = 1 \), the term is \( b_1 = 1^2 + 4 = 5 \).
• As \( n \, \) increases, we observe patterns and trends, helping us to predict future terms.

Sequence evaluation is vital in both understanding and applying the formula that generates the sequence terms. It allows us to systematically calculate each subsequent term by replacing \( n \) with the desired position number.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference. An arithmetic sequence can be expressed in the form: \( a, a + d, a + 2d, \ldots \), where \( a \) is the first term and \( d \) is the common difference.
Unlike our given sequence \( b_n = n^2 + 4 \), which is based on a quadratic function and does not have a constant difference between terms, an arithmetic sequence keeps a uniform change.
A simple example of an arithmetic sequence is the series \( 2, 4, 6, 8, \ldots \) with a common difference of 2. In contrast, the sequence \( 4, 5, 8, 13, 20, \ldots \) from our quadratic example, does not yield any single arithmetic progression as the difference between terms changes as follows: 1, 3, 5, 7.