The ending balances in Olivia’s savings account for each of the past four years form the sequence 800, 850, 900, 950. Is the sequence arithmetic?
In this question, we’re given a sequence of four numbers. And we’re told that the sequence represents the ending balances in Olivia’s savings account over the last four years. And the question wants us to determine if this sequence is an arithmetic sequence. To do this, we’re first going to need to recall what we mean when we say an arithmetic sequence. We recall an arithmetic sequence is a sequence where the difference between any two consecutive terms of our sequence is the same. In other words, to check that a given sequence is an arithmetic sequence, we need to check where the difference between any two consecutive terms remains the same.
And in this question, we want to do this for the sequence of four numbers representing the balance of Olivia’s saving account over four years. Now, we want to check where the difference between any two consecutive terms in this sequence is the same. So we’re going to need to start by calculating the difference between two consecutive terms. Let’s check the first two terms. So let’s start by calculating the difference of the second term and the first term. For this sequence, that’s going to be 850 minus 800. And we can calculate this value; it’s equal to 50. What this tells us is we need to add 50 on to our first term to get to our second term. We can do exactly the same for our third term and second term. We want to calculate the difference between these. So we calculate the third term minus the second term.
The third term in the sequence is 900, and the second term is 850. So we want to calculate 900 minus 850. And if we calculate this expression, we see it’s also equal to 50. And once again, what this is telling us is we need to add 50 on to our second term to reach our third term. And we can do exactly the same for our fourth term and our third term. We’re going to want to calculate the difference between these two terms. The fourth term in the sequence is 950, and the third term is 900. So the difference between these is 950 minus 900, which we can calculate is also equal to 50. And once again, what this tells us is, to get from the third term in our sequence to the fourth term in our sequence, we need to add 50.
And now we can see something interesting. There’s no fifth term in our sequence. So in actual fact, we’ve now calculated all of the differences between consecutive terms of the sequence. And we can see this is always equal to the same value of 50. Therefore, we’ve shown for the sequence given to us in the question, the difference between any two consecutive terms is always equal to 50. Therefore, we can conclude that yes, the sequence given to us in the question is an arithmetic sequence because the difference between any two consecutive terms of this sequence is equal to 50.