Question

given the following arithmetic sequence: 2, 4, 6, 8, 10, ... what is the common difference? 2 what is the value of the first term? 2 write a simplified expression for the right half of the equation that can be used to find the nth term in the sequence. a_n =

Explanation:

Step1: Recall the formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step2: Identify \( a_1 \) and \( d \)

From the sequence \( 2, 4, 6, 8, 10, \dots \), the first term \( a_1 = 2 \) and the common difference \( d=2 \) (since \( 4 - 2 = 2 \), \( 6 - 4 = 2 \), etc.).

Step3: Substitute \( a_1 \) and \( d \) into the formula

Substitute \( a_1 = 2 \) and \( d = 2 \) into \( a_n=a_1+(n - 1)d \):
\[

$$\begin{align*} a_n&=2+(n - 1)\times2\\ &=2 + 2n-2\\ &=2n \end{align*}$$

\]

Answer:

\( 2n \)