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.

Question

Suppose that

{a 1, a 2, a 3, \dots}

is an arithmetic sequence with common difference d. Explain why

{a 1, a 3, a 5, \dots}

is also an arithmetic sequence.

Answer 1

Step 1 It is given that the given sequence is arithmetic sequence with common difference d. The given sequence is: ${a_{1}, a_{2}, a_{3}, \dots}$ where, $a^{1}$ is the first term and d is the common difference Step 2 So, the nth term can be written as: $a_{n} = a + (n - 1) d$ 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. ${a_{1}, a_{3}, a_{5},}$ Step 4 Consider the difference between first and second terms of the above sequence: $a_{3} - a_{1} = (a_{1} + 2 d) - a_{1} [can be written from the n^{(t h)} term of given series] = 2 d$ Step 5 Now, consider the difference between second and third term: $a_{5} - a_{3} = (a_{1} + 4 d) - (a_{1} + 2 d)$
$= 4 d = 2 d$
$= 2 d$ Step 6 Hence the common difference between the terms is same and equal to $2 d$ . Therefore, the series is arithmetic with the common difference as $2 d$ .

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.

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