Question

What are the explicit equation and domain for an arithmetic sequence with a first term of 6 and a second term of 2?

an = 6 − 2(n − 1); all integers where n ≥ 1
an = 6 − 2(n − 1); all integers where n ≥ 0
an = 6 − 4(n − 1); all integers where n ≥ 0
an = 6 − 4(n − 1); all integers where n ≥ 1

Answer 1

A sequence is "arithmetic", if the difference between 2 consecutive terms is always equal.

That is, a sequence
`(a_n)` is arithmetic if:


`a_2-a_1=a_3-a_2=a_4-a_3=.....d`, where d is called the "common difference".
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Consider the sequence with a first term of 6 and a second term of 2

the common difference d is
`a_2-a_1=2-6=-4`,

so we construct the sequence as follows:


`a_1=6`


`a_2=6+(-4)=2`


`a_3=[6+(-4)]+(-4)=-2`


`a_4=[6+(-4)+(-4)]+(-4)=-6`


`a_5=[6+(-4)+(-4)+(-4)]+(-4)=-10`

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thus clearly
`a_n=[6+(-4)(n-1)=a_n=[6-4(n-1)`

and n∈{1, 2, 3, 4...}


Answer:

an = 6 − 4(n − 1); all integers where n ≥ 1