Problem 8 Find the terms $a_{0}, a_{1},$... [FREE SOLUTION]

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Precalculus : Building Concepts and Connections

Revathi Narasimhan

$Math Studyset 91影视 Explanations$ Math

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Chapter 10: Problem 8

Find the terms $a_{0}, a_{1},$ and $a_{2}$ for each sequence. $$a_{n}=-3+2 n$$

Short Answer

Expert verified

The first three terms of the sequence are $a_{0} = -3$, $a_{1} = -1$, and $a_{2} = 1$.

Step by step solution

Find the first term

To find the first term, $a_{0}$, we substitute $n=0$ into the sequence formula. Thus, $a_{0} = -3 + 2*0 = -3$.

Find the second term

To find the second term, $a_{1}$, we substitute $n=1$ into the sequence formula. Thus, $a_{1} = -3 + 2*1 = -1$.

Find the third term

To find the third term, $a_{2}$, we substitute $n=2$ into the sequence formula. Thus, $a_{2} = -3 + 2*2 = 1 $.

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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 that represents the general rule for finding any term in a sequence. For arithmetic sequences, the formula typically includes a starting value and a constant difference. In the original exercise, the sequence formula given is $a_{n} = -3 + 2n$. This formula shows:

The sequence starts with the term $-3$, known as the initial term when $n = 0$.
The number $2$ represents the common difference, which is the amount added to each term to get the next term in the sequence.

To find any term, $a_{n}$, in this sequence, we substitute the desired term position $n$ into the formula to calculate the term value. Understanding the formula enables easy prediction of future terms without listing all preceding numbers.

Terms of a Sequence

The terms of a sequence are the individual elements that make up the sequence. Each term in a sequence corresponds to a specific position, denoted by $n$. In the original exercise, the aim was to find the first three terms: $a_{0}$, $a_{1}$, and $a_{2}$. The terms are:

$a_{0} = -3$
$a_{1} = -1$
$a_{2} = 1$

These terms are determined by substituting specific values of $n$ (such as 0, 1, and 2) into the sequence formula. Each substitution provides a numerical value, representing the term's position in the sequence. Recognizing terms and how they relate to the sequence formula is crucial for identifying patterns and understanding the progression of sequences.

Substitution in Sequences

Substitution in sequences involves replacing the variable $n$ in the sequence formula with a specific number to find the corresponding term. This process allows us to calculate exact term values by plugging in the position number. Here's how it works using the original exercise:

To find $a_{0}$, substitute $n = 0$ into the formula: $a_{0} = -3 + 2\times0 = -3$.
To find $a_{1}$, substitute $n = 1$ into the formula: $a_{1} = -3 + 2\times1 = -1$.
To find $a_{2}$, substitute $n = 2$ into the formula: $a_{2} = -3 + 2\times2 = 1$.

Substituting different values of $n$ enables you to find any term in the sequence quickly and accurately. Understanding substitution helps demystify sequences, making them more accessible and easier to work with.

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91影视

Find the terms \(a_{0}, a_{1},\) and \(a_{2}\) for each sequence. $$a_{n}=-3+2 n$$

Short Answer

Step by step solution

Find the first term

Find the second term

Find the third term

Key Concepts

Sequence Formula

Terms of a Sequence

Substitution in Sequences

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