# Suppose that left{a1, a2, a3, cdotsright} is an arithmetic sequence with common difference d. Explain why left{a1, a3, a5, cdotsright} is also an arithmetic sequence.

Suppose that is an arithmetic sequence with common difference d. Explain why is also an arithmetic sequence.
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Brittany Patton

Step 1 It is given that the given sequence is arithmetic sequence with common difference d. The given sequence is: where, ${a}^{1}$ is the first term and d is the common difference Step 2 So, the nth term can be written as: Step 3 Now, it is required to prove that the below sequence is also arithmetic that is the common difference is same between each consecutive terms. Step 4 Consider the difference between first and second terms of the above sequence: Step 5 Now, consider the difference between second and third term:
$=4d=2d$
$=2d$ Step 6 Hence the common difference between the terms is same and equal to $2d$. Therefore, the series is arithmetic with the common difference as $2d$.

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