Question

Terms of an Arithmetic Sequence Determine the common difference, the fifth term, the $n\text{th}$ term, and the $100\text{th}$ term of the arithmetic sequence.

$29,11,-7,-25,\dots$

Short answer

The common difference is $-18$, the fifth term is $-43$, the $n\text{th}$ term is $a_n=29-18\left(n-1\right)$ and the $100\text{th}$ term is $-1753$.

Step-by-step solution

Step 1. Given information.

The given arithmetic sequence is:

$29,11,-7,-25,\dots$

Step 2. Write the concept.

The $n\text{th}$ term of an arithmetic sequence is:

$a_n=a+\left(n-1\right)d$

Where $\mathit{a}$ is the first term and $d$is a common difference.

Step 3. Determine the common difference.

Subtract the first term from the second term of the arithmetic sequence to find the common difference.

$\begin{array}{c}d=a_2-a_1\\ =11-29\\ =-18\end{array}$

The common difference of the given arithmetic sequence is$-18.$

Step 4. Determine the fifth term.

Add the common difference in the fourth term to find the fifth term.

$\begin{array}{c}{a}_5=a_4+6\\ =-25+\left(-18\right)\\ =-25-18\\ =-43\end{array}$

The fifth term of the given arithmetic sequence is$-43.$

Step 5. Determine the nth term.

Substitute $a=29$and $d=-18$in $a_n=a+\left(n-1\right)d$to find the $n\text{th}$term.

$\begin{array}{c}{a}_n=\left(29\right)+\left(n-1\right)\left(-18\right)\\ a_n=29-18\left(n-1\right)\end{array}$

The term of the given arithmetic sequence is

$a_n=29-18\left(n-1\right)$

Step 6. Determine the 100th term.

Substitute $n=100$in $a_n=29-18\left(n-1\right)$to find the $100\text{th}$term.

$\begin{array}{c}{a}_{100}=29-18\left(100-1\right)\\ =29-18\left(99\right)\\ =29-1782\\ =-1753\end{array}$

Thus, the common difference is $-18$, the fifth term is $-43$, the $n\text{th}$ term is $a_n=29-18\left(n-1\right)$ and the $100\text{th}$ term is $-1753$.