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Chapter 14: Problem 6

$$ \begin{aligned} &a_{n}=2 n^{2}-6 \\ &-4,2,12,26,44 \end{aligned} $$

Short Answer

Expert verified
The given sequence matches the formula \(a_n = 2n^2 - 6\).

Step by step solution

01

Understanding the Sequence Formula

The sequence is given by the formula \( a_n = 2n^2 - 6 \). This formula tells us that for any term \( a_n \), we need to substitute the value of \( n \) into the formula to find the corresponding sequence term.
02

Substituting Values to Find Terms

Let's find the terms for \( n = 1, 2, 3, 4, \) and \( 5 \): - For \( n = 1 \), \( a_1 = 2(1)^2 - 6 = 2 - 6 = -4 \).- For \( n = 2 \), \( a_2 = 2(2)^2 - 6 = 8 - 6 = 2 \).- For \( n = 3 \), \( a_3 = 2(3)^2 - 6 = 18 - 6 = 12 \).- For \( n = 4 \), \( a_4 = 2(4)^2 - 6 = 32 - 6 = 26 \).- For \( n = 5 \), \( a_5 = 2(5)^2 - 6 = 50 - 6 = 44 \).
03

Comparing with Given Sequence

The terms we calculated \(-4, 2, 12, 26, 44\) match the sequence given in the problem \(-4, 2, 12, 26, 44\). Therefore, the formula is verified to generate the sequence.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Formula
A sequence formula is a mathematical expression defining the rule for generating the terms of a sequence. In the given exercise, the sequence is defined by the formula \( a_n = 2n^2 - 6 \). This indicates that the sequence follows a quadratic pattern.Here鈥檚 how a sequence formula works:
  • Each term in the sequence is the result of applying the formula with a particular value of \( n \), the term's position in the sequence.
  • The sequence starts at \( n = 1 \) and continues with consecutive natural numbers \( n = 2, 3, 4, \ldots \).
  • The formula assists in predicting any term of the sequence without directly counting each preceding term.
Sequence formulas, such as the quadratic sequence in this problem, are commonly used in mathematics to describe patterns and relationships through predictable, mathematical rules.
Term Calculation
In mathematics, calculating each term of a sequence involves evaluating the sequence formula for specific values of \( n \). Let's discuss how term calculation works for the problem at hand.Using the formula \( a_n = 2n^2 - 6 \):
  • Identify the position \( n \) of the term you want to calculate. For example, \( n = 3 \) corresponds to the third term.
  • Substitute the value of \( n \) into the formula: for \( n = 3 \), compute \( a_3 = 2(3)^2 - 6 \).
  • Perform the arithmetic operations: calculate \( 2(3)^2 = 2 \times 9 = 18 \), then subtract 6 to get 12. Hence, the third term \( a_3 \) is 12.
By following these steps for different values of \( n \), you can determine each term in the sequence accurately. This process showcases how mathematical formulas enable precise calculations of sequence terms.
Substitution Method
The substitution method is a simple yet powerful technique used in sequences for calculating terms. It involves replacing variables in a formula with specific numerical values.Here's how substitution is applied:
  • Identify the sequence formula, which in our case is \( a_n = 2n^2 - 6 \).
  • Select the value of \( n \), which represents the term position you are interested in. For instance, to find the fourth term choose \( n = 4 \).
  • Substitute \( n \) with the chosen value in the formula: \( a_4 = 2(4)^2 - 6 \).
  • Evaluate the arithmetic expressions: \( 2(4)^2 = 2 \times 16 = 32 \), and then subtract 6 leading to \( 32 - 6 = 26 \).
Substitution transforms a general formula into a specific numeric calculation, thus illuminating each term in the sequence. It is a handy tool for efficiently determining sequence values, especially in problems involving quadratic sequences.

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