The sequence 𝑎 𝑛 is an arithmetic sequence if 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛 is equal to what.
In this question, we’re given a sequence 𝑎 𝑛. And we need to determine the property that 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛 will have, which will make this an arithmetic sequence. That’s the difference between consecutive terms. To do this, let’s start by recalling what we mean by an arithmetic sequence. We say that a sequence is an arithmetic sequence if the difference between any two consecutive terms in our sequence is constant. And another way of thinking about this is saying we add a constant value to the previous term to generate the next term. For example, in this sequence, we add four to generate the next term. And we’re almost ready to answer this question.
However, let’s first recall what we mean by the notation 𝑎 sub 𝑛 plus one and 𝑎 sub 𝑛. The notation 𝑎 sub 𝑛 means the value of the 𝑛th term in the sequence. For example, if we call our sequence one, five, nine, 13, 17, 𝑎 sub 𝑛, then the second term in our sequence is five. So 𝑎 sub two is equal to five. Similarly, 𝑎 sub one is equal to one, since the value of the first term is one. This means the expression 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛 is the difference between consecutive terms in our sequence.
For example, in our example of an arithmetic sequence, one difference of successive terms is the second term minus the first term, which is five minus one. And this value is equal to four, because we found 𝑎 sub two by adding four to 𝑎 sub one. But this will be true for any two consecutive terms in our sequence, since the next term in our sequence is the previous term plus four. This will hold true for any arithmetic sequence.
Any arithmetic sequence is one where the difference between consecutive terms is constant. And another way of saying this is 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛 is constant for any integer value of 𝑛 greater than or equal to one.