Problem 17

Question

Find $a_{8}$ and $a_{n}$ for each arithmetic sequence. $$a_{1}=8, a_{2}=6$$

Step-by-Step Solution

Verified

Answer

$a_8 = -6$, $a_n = 10 - 2n$.

1Step 1: Identify common difference

To find the common difference ($d$) of the arithmetic sequence, subtract the first term from the second term. From the problem, $a_1 = 8$ and $a_2 = 6$. Calculate $d = a_2 - a_1 = 6 - 8 = -2$.

2Step 2: Use the explicit formula

The explicit formula for the $n$-th term of an arithmetic sequence is $a_n = a_1 + (n-1)d$. Here, $a_1 = 8$ and $d = -2$. Plug these values into the formula: $a_n = 8 + (n-1)(-2)$.

3Step 3: Simplify the formula

Simplify the explicit formula: $a_n = 8 + (n-1)(-2) = 8 - 2(n-1) = 8 - 2n + 2 = 10 - 2n$. Thus, $a_n = 10 - 2n$.

4Step 4: Calculate $a_{8}$

Substitute $n = 8$ into the simplified formula: $a_8 = 10 - 2(8) = 10 - 16 = -6$. So, $a_8 = -6$.

Key Concepts

Understanding Common DifferenceExplicit Formula of Arithmetic SequencesCalculating the n-th Term

Understanding Common Difference

The common difference in an arithmetic sequence represents the constant amount that each term in the sequence differs from the previous term. This is a fundamental aspect of arithmetic sequences, which distinguish them from other types of sequences. To find the common difference ($d$) in a sequence, you simply subtract the first term from the second term.

In our example, the arithmetic sequence started with the terms $a_1 = 8$ and $a_2 = 6$. By performing the operation $a_2 - a_1$, we calculated:

$d = 6 - 8 = -2$.

This result tells you that each term decreases by 2 from one term to the next. Knowing and applying the common difference allows us to find any term in the sequence.

Explicit Formula of Arithmetic Sequences

The explicit formula is a crucial formula that allows us to find the $n$-th term of an arithmetic sequence without having to calculate all of the previous terms. This formula is expressed as follows:
$a_n = a_1 + (n-1)d$
In this formula:

$a_n$ represents the $n$-th term you want to find.
$a_1$ is the first term of the sequence.
$n$ denotes the term number.
$d$ is the common difference.

Using the given values from the example where $a_1 = 8$ and $d = -2$, we plug these into the explicit formula:
$a_n = 8 + (n-1)(-2)$.

Simplifying leads us to the function:
$a_n = 10 - 2n$.
This handy formula allows you to find the value of any term in the sequence efficiently.

Calculating the n-th Term

Calculating the $n$-th term involves using the simplified explicit formula to find specific terms in an arithmetic sequence. If you know the first term and the common difference, you can easily compute any term position. For the sequence in our problem, once we simplified the formula to $a_n = 10 - 2n$, calculating the eighth term $a_8$ becomes straightforward.
To find $a_8$, substitute $n = 8$ into the formula:

$a_8 = 10 - 2(8)$
$a_8 = 10 - 16$
$a_8 = -6$

Thus, the 8th term of the sequence is $-6$. This method streamlines the process of finding terms without needing to calculate each sequentially, saving time and effort.

Problem 17

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