Problem 14 - Algebra for College Students

Question

Find the 10th and 15th terms of the sequence where \(a_{n}=-n^{2}-10 . \quad a_{10}=-110, a_{15}=-235\)

Step-by-Step Solution

Answer

The 10th term is -110 and the 15th term is -235.

Step 1: Identify the Formula of the Sequence

The formula for the sequence is given as \(a_{n} = -n^{2} - 10\). This formula can be used to find any term in the sequence by substituting the term number \(n\) into the formula.

Step 2: Calculate the 10th Term

To find the 10th term of the sequence, substitute \(n = 10\) into the formula: \(a_{10} = -(10)^2 - 10 = -100 - 10 = -110\).

Step 3: Calculate the 15th Term

To find the 15th term of the sequence, substitute \(n = 15\) into the formula: \(a_{15} = -(15)^2 - 10 = -225 - 10 = -235\).

Key Concepts

Quadratic sequences

In algebra, a quadratic sequence is a type of sequence where the difference between terms isn't constant, but the differences of the differences (the second differences) are constant. Quadratic sequences are related to quadratic functions, which are expressed as a polynomial of degree two.
The general formula for a quadratic sequence is given by the equation:

• \( a_n = an^2 + bn + c \)

where \( a, b, \) and \( c \) are constants, and \( n \) represents the term number.
In the context of the exercise, the sequence formula is \( a_n = -n^2 - 10 \). Here, \( a = -1 \), \( b = 0 \), and \( c = -10 \). This indicates that the sequence is indeed quadratic because its highest term is \( n^2 \).
One key feature of quadratic sequences is that they graph as a parabola when plotted, either opening upwards or downwards depending on the sign of the quadratic term, which in this case is negative, indicating a downward opening parabola.

Finding terms in a sequence

To find any term in a sequence, you substitute the position number into the sequence's formula. This process requires understanding the role of the term number in the formula.
In our example, to find the 10th term, you replace \( n \) with 10:

• \( a_{10} = -(10)^2 - 10 = -110 \)

Similarly, to find the 15th term, substitute \( n \) with 15:

• \( a_{15} = -(15)^2 - 10 = -235 \)

This method follows because the formula \( a_n = -n^2 - 10 \) gives a rule that is applied to any term position \( n \) to produce the sequence value at that position. Each computation involves simple arithmetic operations after substituting the term number into the formula.
By utilizing the formula efficiently, you can find any desired term in the sequence without having to manually calculate the values of all preceding terms, saving time and effort.

Algebraic expressions

Algebraic expressions are combinations of letters (variables), numbers, and arithmetic operations. They represent quantities or relationships using symbols and are fundamental in forming equations and functions.
When managing sequences, algebraic expressions define the rule or formula for generating terms of the sequence. In the case of the quadratic sequence in the exercise, the expression \( -n^2 - 10 \) determines each term's value based on its position.
These expressions sometimes involve variables raised to a power, constants, and coefficients. In our example:

• The expression is \(-n^2 - 10\), where \(-n^2\) represents the quadratic term, and \(-10\) is a constant.

Algebraic expressions allow us to succinctly represent complex ideas in a manageable way, enabling us to predict, calculate, and analyze a wide range of mathematical scenarios. Working with these expressions, we can easily and efficiently determine the characteristics and values within sequences.