Question Video: Finding the Common Difference and General Term of an Arithmetic Sequence Mathematics

In any sequence pattern, if the difference between any two successive terms is a fixed number, 𝑑, then this is an arithmetic sequence. Consider the sequence 1, 4, 7, 10, and β‹― , and then answer the following questions. Is the sequence arithmetic? What is the value of 𝑑? What is the general term of this sequence with 𝑛 β‰₯ 0?

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Video Transcript

In any sequence pattern, if the difference between any two successive terms is a fixed number 𝑑, then this is an arithmetic sequence. Consider the sequence one, four, seven, 10, and so on and then answer the following questions. Is the sequence arithmetic? What is the value of 𝑑? What is the general term of this sequence with 𝑛 is greater than or equal to zero?

To start this question, we’re given a little reminder about the definition of an arithmetic sequence. And it’s one which has a difference or common difference between any two successive terms the same. This word successive is like the word consecutive. It simply means two terms where one immediately follows the other one. We are asked to consider the sequence one, four, seven, 10, and so on. If we look at the first question, we’re asked if the sequence is arithmetic. So if it’s arithmetic, it will have a common difference between any two consecutive terms.

So if we wanted to find the difference between the first and the second term, we would work out four take away one, which of course is three. To find the next common difference between the third term and the second term, we would work out seven take away four, and that’s also three. In the same way, the difference between 10 and seven is also three. As we have a common difference, then we do have an arithmetic sequence. And we can say yes as the answer for the first part of this question. The second part of the question asks us, what is the value of 𝑑? 𝑑 is the common difference, and we’re reminded of that in the question text. And it’s also nice and easy to work out; it’s three. And that’s the second part of the question answered.

In the final part of this question, we’re asked to find the general term of this sequence with 𝑛 greater than or equal to zero. Remember that the general term is often seen as the 𝑛th term. The fact that 𝑛 is greater than or equal to zero means that the index of this sequence should start with 𝑛 equals zero. So actually, this sequence begins with a zeroth term. Then we’d have the first term, then the second term and the third term and so on. We should recall that there is a formula to help us find the 𝑛th term of an arithmetic sequence. The 𝑛th term π‘Ž sub 𝑛 is equal to π‘Ž plus 𝑛 minus one multiplied by 𝑑, where π‘Ž is the first term and 𝑑 is the common difference.

When we look at the sequence one, four, seven, 10, and so on, the first term is really the same as π‘Ž sub one. So the value of π‘Ž, which we plug in, will be four. And as the value of 𝑑, which we worked out earlier, is three, then we add on 𝑛 minus one multiplied by three. When we distribute the three across the parentheses, we get three multiplied by 𝑛, which is three 𝑛, and three multiplied by negative one, which is negative three. Finally, when we simplify this, we get four minus three, which is one, plus three 𝑛 or three 𝑛 plus one. And that’s the answer for the third part of this question: the general term or 𝑛th term of this sequence is three 𝑛 plus one.

But it is important to note that this was because the index 𝑛 was greater than or equal to zero. If the index had started with one, i.e., 𝑛 is greater than or equal to one, then we would have worked out the 𝑛th term to be three 𝑛 minus two. So it’s really important to read the question to see if there’s an indication that the index, in fact, begins with zero. In this case, however, we can give the general term as three 𝑛 plus one.

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