Question

In Exercises \(12-15\), write the first 5 terms of the sequence \(a_{n}=f(n)\) $$ f(x)=x^{2}+1 $$

Short answer

The first 5 terms of the sequence are 2, 5, 10, 17, and 26.

Step-by-step solution

Write the general formula for the sequence

The general formula of the sequence is given as: $$ a_n = f(n) = n^2 + 1 $$

Calculate the first term, \(a_1\)

Plug in \(n=1\) into the formula $$ a_1 = f(1) = (1^2) + 1 = 1 + 1 = 2 $$

Calculate the second term, \(a_2\)

Plug in \(n=2\) into the formula $$ a_2 = f(2) = (2^2) + 1 = 4 + 1 = 5 $$

Calculate the third term, \(a_3\)

Plug in \(n=3\) into the formula $$ a_3 = f(3) = (3^2) + 1 = 9 + 1 = 10 $$

Calculate the fourth term, \(a_4\)

Plug in \(n=4\) into the formula $$ a_4 = f(4) = (4^2) + 1 = 16 + 1 = 17 $$

Calculate the fifth term, \(a_5\)

Plug in \(n=5\) into the formula $$ a_5 = f(5) = (5^2) + 1 = 25 + 1 = 26 $$ The first 5 terms of the sequence are: 2, 5, 10, 17, and 26.

Key concepts

Quadratic Functions

Quadratic functions are an essential element of mathematics and appear in many forms and situations. They are polynomial functions of degree two, generally represented as \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In this exercise, the quadratic function provided is \(f(x) = x^2 + 1\). This specific function is simple because it lacks the linear term (\(bx\)).

For visualizing quadratic functions, they graph as parabolas. All parabolas have a symmetric "U" shape, which can open upwards or downwards, depending on the sign of \(a\). In our case, the graph opens upwards, as the coefficient of \(x^2\) is positive (1).

Understanding the nature of quadratic functions is crucial when dealing with sequences or any related math problem involving polynomials.

Sequence Terms

Sequence terms are essentially the individual elements or numbers that form a sequence. When we describe a sequence mathematically, we often use a formula based on \(n\), the position of a term.

For this exercise, we can use the quadratic function to define a sequence: \(a_n = n^2 + 1\). Each term in the sequence is calculated by plugging in successive integer values for \(n\). So, for the first five terms:

• First term (\(a_1\)): Plug in \(n = 1\), so \(a_1 = (1^2) + 1 = 2\).
• Second term (\(a_2\)): Plug in \(n = 2\), so \(a_2 = (2^2) + 1 = 5\).
• Third term (\(a_3\)): Plug in \(n = 3\), so \(a_3 = (3^2) + 1 = 10\).
• Fourth term (\(a_4\)): Plug in \(n = 4\), so \(a_4 = (4^2) + 1 = 17\).
• Fifth term (\(a_5\)): Plug in \(n = 5\), so \(a_5 = (5^2) + 1 = 26\).

This results in the sequence: 2, 5, 10, 17, and 26. Recognizing how to find sequence terms from a formula can greatly simplify working with sequences in mathematics.

Step-by-step Calculation

Step-by-step calculation is a methodical process where each term of a sequence is computed individually. This approach ensures clarity and precision, particularly for those new to the concept.

In this specific problem, we start by identifying the formula: \(a_n = n^2 + 1\). For each step, substitute the specific term number into the equation:

• Step 1: Calculate \(a_1\) by replacing \(n\) with 1.
• Step 2: Calculate \(a_2\) by replacing \(n\) with 2.
• Step 3: Calculate \(a_3\) by replacing \(n\) with 3.
• Step 4: Calculate \(a_4\) by replacing \(n\) with 4.
• Step 5: Calculate \(a_5\) by replacing \(n\) with 5.

By following this logical and organized method, each term is verified independently, preventing errors and reinforcing the understanding of how sequences are built from mathematical formulas. This technique is particularly beneficial in educational settings to build confidence and mastery over mathematical concepts.