Question

$n\text{th}$ Term of an Arithmetic sequence Find the $n\text{th}$ term of the arithmetic sequence with given first term $a$ and common difference $d$. What is the $10\text{th}$ term?

$a=\sqrt{3},d=\sqrt{3}$

Short answer

The $n\text{th}$ term of the arithmetic sequence is $a_n=\sqrt{3}n$ and the $10\text{th}$ term of the sequence is $10\sqrt{3}$.

Step-by-step solution

Step 1. Given information.

The first term is $a=\sqrt{3}$ and the common difference is $d=\sqrt{3}$.

Step 2. Write the concept.

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

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

Where $a$ is the first term and $d$ is a common difference.

Step 3. Determine the nth term.

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

$\begin{array}{l}{a}_n=\sqrt{3}+\left(n-1\right)\left(\sqrt{3}\right)\\ a_n=\sqrt{3}+\sqrt{3}n-\sqrt{3}\\ a_n=\sqrt{3}n\end{array}$

Step 4. Determine the 10th term.

Substitute $n=10$in $a_n=\sqrt{3}n$to find the $10\text{th}$term of the sequence.

$\begin{array}{l}{a}_{10}=\sqrt{3}\left(10\right)\\ =10\sqrt{3}\end{array}$

Thus, the $n\text{th}$ term of the arithmetic sequence is $a_n=\sqrt{3}n$ and the $10\text{th}$ term of the sequence is $10\sqrt{3}$.