Problem 45

Question

For the following exercises, write an explicit formula for each arithmetic sequence. $$ a=\\{-5,95,195, \ldots\\} $$

Step-by-Step Solution

Verified

Answer

The explicit formula for the sequence is $ a_n = 100n - 105 $.

1Step 1: Identify the First Term

The first term of the sequence is given as $ a_1 = -5 $. This will serve as the starting point for our explicit formula.

2Step 2: Calculate the Common Difference

Subtract the first term from the second term to find the common difference. The second term is $95$, so we have $ d = 95 - (-5) = 100 $. Thus, the common difference is $100$.

3Step 3: Write the General Formula for an Arithmetic Sequence

The general formula for an arithmetic sequence is $ a_n = a_1 + (n-1) \cdot d $. Here, $ a_1 = -5 $ and $ d = 100 $. Substitute these values into the formula: $ a_n = -5 + (n-1) \cdot 100 $.

4Step 4: Simplify the Formula

Simplify the formula $ a_n = -5 + (n-1) \cdot 100 $ to get $ a_n = -5 + 100n - 100 $. This simplifies further to $ a_n = 100n - 105 $ by combining like terms.

Key Concepts

Explicit FormulaCommon DifferenceFirst TermGeneral Formula

Explicit Formula

An explicit formula in an arithmetic sequence provides a robust method for determining any term in the sequence without computing all the preceding terms. Consider it as a mathematical shortcut to arrive at the desired point directly. The basic structure of this formula comes from the pattern of adding a constant (the common difference) repeatedly.Let's look at our sequence: $-5, 95, 195,\ldots$. We crafted the explicit formula for this arithmetic sequence as $a_n = 100n - 105$. This formula starts from the constant linear progression of the sequence.

Here, $n$ represents the position of the term you want to find.
Substitute $n$ with the term number in the formula, and out comes your specific sequence term.

For instance, if you want the 3rd term, set $n = 3$ in the formula and calculate.

Common Difference

The common difference in an arithmetic sequence is the value you add each time to get from one term to the next. It's the heartbeat of the sequence’s rhythm.To find it, take any term in the sequence and subtract its previous term from it. In our example sequence $-5, 95, 195,\ldots$, we found the common difference by subtracting the first term $-5$ from the second term $95$. The calculation is: \[d = 95 - (-5) = 100\]Therefore, our common difference $d$ is $100$.

This regular amount, $100$, is what gets added to each successive term.
It simplifies tracking, since each term grows uniformly by $100$.

Knowing the common difference allows you to predict what comes next in an arithmetic sequence easily.

First Term

In arithmetic sequences, the first term is the starting point, laying the foundation for all subsequent terms. It is often denoted by $a_1$ and it marks the origin of the sequence.In the sequence given, the first term, $a_1$, is $-5$. This means our sequence starts at $-5$, from which every other term is calculated by adding the common difference, $100$, repeatedly.

Knowing the first term is essential because it sets the initial condition of the sequence.
All calculations of future terms stem from this origin.

With the first term and the common difference, you're well-equipped to drive through the rest of the sequence by applying them in the explicit formula.

General Formula

The general formula for an arithmetic sequence is your mainstay tool. It crafts the pathway for you to find any term in the sequence.This formula takes the basic form: \[a_n = a_1 + (n-1) \cdot d\]Where:

$a_n$ is the $n$th term you're targeting.
$a_1$ is the first term, which is $-5$ in this case.
$d$ is the common difference, which is $100$.
$n$ is the term number.

By substituting $-5$ for $a_1$ and $100$ for $d$, you derive our specific formula: \[a_n = -5 + (n-1) \cdot 100\]This simplifies further to: \[a_n = 100n - 105\]You can use this formula to pinpoint the value of any term by plugging different values of $n$ into it.

Problem 45

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