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.

Braxton Pugh 2020-11-09 Answered
Suppose that {a1, a2, a3, } is an arithmetic sequence with common difference d. Explain why {a1, a3, a5, } is also an arithmetic sequence.
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Expert Answer

Brittany Patton
Answered 2020-11-10 Author has 100 answers

Step 1 It is given that the given sequence is arithmetic sequence with common difference d. The given sequence is: {a1, a2, a3, } where, a1 is the first term and d is the common difference Step 2 So, the nth term can be written as: an=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. {a1, a3, a5,} Step 4 Consider the difference between first and second terms of the above sequence: a3  a1=(a1 + 2d)  a1 [can be written from the n(th) term of given series]=2d Step 5 Now, consider the difference between second and third term: a5  a3=(a1 + 4d)  (a1 + 2d)
=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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