Lesson Video: Arithmetic Series Mathematics

Video Transcript

In this video, we will learn how to calculate the sum of the terms in an arithmetic sequence with a definite number of terms. We will begin by recalling what we mean by an arithmetic sequence and arithmetic series.

The list of numbers written in a definite order is called a sequence. In an arithmetic sequence, the difference between one term and the next is a constant. This is known as the common difference and is denoted by the letter 𝑑. The first term of an arithmetic sequence 𝑎 sub one is usually just written as 𝑎. The second term 𝑎 sub two will therefore be equal to 𝑎 plus 𝑑. To get to the third term, we will need to add 𝑑 again, so 𝑎 sub three is equal to 𝑎 plus 𝑑 plus 𝑑. This can be simplified to 𝑎 plus two 𝑑. This pattern continues so that the fourth term is 𝑎 plus three 𝑑, the fifth term 𝑎 plus four 𝑑, and so on.

We notice that the second term has one 𝑑. The third term has two 𝑑’s. The fourth term would have three 𝑑’s and so on. This means that the 𝑛th term would have 𝑛 minus one 𝑑’s. This leads us to the formula for the general term of an arithmetic sequence 𝑎 sub 𝑛 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑. When dealing with a finite sequence, as in this video, we can denote the last term as 𝑙. The sum of terms of an arithmetic sequence is called an arithmetic series. This means that an arithmetic series would be written in the form 𝑎 plus 𝑎 plus 𝑑 plus 𝑎 plus two 𝑑 and so on all the way up to 𝑎 plus 𝑛 minus one multiplied by 𝑑.

We will now use this information to prove two formulae to calculate the sum of the first 𝑛 terms of an arithmetic sequence.

Find an expression for the sum of an arithmetic sequence whose first term is 𝑎 and whose common difference is 𝑑.

We are told in the question that the first term of our arithmetic sequence is 𝑎 and the common difference is 𝑑. We are trying to find an expression for the sum of the first 𝑛 terms which we will write as 𝑆 sub 𝑛. This will be equal to the first term 𝑎 plus the second term 𝑎 plus 𝑑 and so on. We know that the 𝑛th term of any arithmetic sequence is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑. This means that the penultimate term is equal to 𝑎 plus 𝑛 minus two multiplied by 𝑑. We will call this equation one.

We will then reverse the order of this sum, which we can do as addition is commutative. This gives us 𝑆 sub 𝑛 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑 plus 𝑎 plus 𝑛 minus two multiplied by 𝑑 and so on and finally plus 𝑎 plus 𝑑 plus 𝑎. We will call this equation two. Adding equation one and equation two gives us two multiplied by 𝑆 sub 𝑛 on the left-hand side. On the right-hand side, we will add each pair of terms. Adding the first pair, we see that 𝑎 plus 𝑎 is equal to two 𝑎. So we have two 𝑎 plus 𝑛 minus one multiplied by 𝑑.

The second pair of terms have the same sum as 𝑎 plus 𝑎 is equal to two 𝑎 and 𝑑 plus 𝑛 minus two 𝑑 is equal to 𝑛 minus one 𝑑. In fact, this will be true for each of the pairs in our equations. Each pair of terms will sum to give us two 𝑎 plus 𝑛 minus one multiplied by 𝑑. We have 𝑛 of these terms, so we can rewrite the right-hand side as 𝑛 multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑. Dividing both sides of our equation by two gives us 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑. This is an expression for the sum of an arithmetic sequence whose first term is 𝑎 and whose common difference is 𝑑.

We will now look at an alternative formula for the sum of an arithmetic sequence.

Write an expression for the sum of the first 𝑛 terms of an arithmetic sequence with first term 𝑎 and last term 𝑙.

We know that the sum of the first 𝑛 terms of an arithmetic sequence can be calculated using the formula 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑, where 𝑛 is the number of terms, 𝑎 is the first term, and 𝑑 is the common difference. We also know that the 𝑛th term 𝑎 sub 𝑛 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑. We are told in this question that 𝑙 is the last term. Therefore, 𝑙 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑. Since 𝑎 plus 𝑎 is equal to two 𝑎, we can rewrite our formula for 𝑆 sub 𝑛 as 𝑛 over two multiplied by 𝑎 plus 𝑎 plus 𝑛 minus one multiplied by 𝑑.

We know that the second part inside the brackets 𝑎 plus 𝑛 minus one multiplied by 𝑑 is equal to 𝑙. The sum of the first 𝑛 terms of an arithmetic sequence with first term 𝑎 and last term 𝑙 is therefore equal to 𝑛 over two multiplied by 𝑎 plus 𝑙.

We will now briefly summarize the formulae we will use for the remainder of this video. The 𝑛th term 𝑎 sub 𝑛 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑. The sum of the first 𝑛 terms written 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑. As we’re dealing with finite sequences, there will be a last term 𝑙. This will be equal to 𝑎 sub 𝑛. Therefore, 𝑆 sub 𝑛 is also equal to 𝑛 over two multiplied by 𝑎 plus 𝑙. We will now use these formulae to solve problems involving finite arithmetic sequences.

Find the sum of the first 10 terms of the arithmetic sequence whose first term is five and common difference is eight.

We are told that the first term, denoted by the letter 𝑎, is equal to five. The common difference 𝑑 of our arithmetic sequence is equal to eight. As we need to calculate the sum of the first 10 terms, 𝑛 is equal to 10. We know that the sum of the first 𝑛 terms of an arithmetic sequence, denoted 𝑆 sub 𝑛, is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑. Substituting in our values of 𝑎, 𝑑, and 𝑛, we can calculate 𝑆 sub 10. This is equal to 10 divided by two multiplied by two multiplied by five plus 10 minus one multiplied by eight. Two multiplied by five is equal to 10, and 10 minus one multiplied by eight is equal to 72. Multiplying 82 by five gives us 410.

The sum of the first 10 terms of the arithmetic sequence whose first term is five and common difference is eight is 410.

We will now look at a similar question where our terms are negative.

Find the sum of the terms of the 11-term arithmetic sequence whose first term is negative 92 and last term is negative 102.

The first term 𝑎 of our arithmetic sequence is equal to negative 92, and the last term 𝑙 is equal to negative 102. We are also told there are 11 terms in the sequence. Therefore, 𝑛 is equal to 11. We could use this information to calculate the common difference 𝑑. However, in this question, it is not required. We can calculate the sum of the first 𝑛 terms using the formula 𝑛 over two multiplied by 𝑎 plus 𝑙. Substituting in our values, we see that 𝑆 sub 11 is equal to 11 over two multiplied by negative 92 plus negative 102. 11 divided by two is equal to 5.5, and negative 92 plus negative 102 is equal to negative 194. Multiplying these two values gives us negative 1067.

The sum of the 11 terms in the arithmetic sequence whose first term is negative 92 and last term is negative 102 is negative 1067.

In our next question, we need to find the sum of the first 10 terms of an arithmetic sequence given the 𝑛th term.

Find the sum of the first 10 terms of the sequence 𝑎 sub 𝑛, where 𝑎 sub 𝑛 is equal to two 𝑛 plus four.

There are a few ways of approaching this problem. One way would be to calculate the first and last terms of the sequence. As there are 10 terms, these are denoted by 𝑎 sub one and 𝑎 sub 10. The first term will be equal to two multiplied by one plus four. This is equal to six. The tenth term 𝑎 sub 10 is equal to two multiplied by 10 plus four. This is equal to 24. We can now use the formula 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by 𝑎 plus 𝑙, where 𝑎 is equal to six, the first term, and 𝑙 is equal to 24, the 10th or last term. 𝑆 sub 10 is equal to 10 over two multiplied by six plus 24. This simplifies to five multiplied by 30, giving us an answer for the sum of the first 10 terms of the sequence of 150.

An alternative method would be to have recognized our sequence is linear. Therefore, the common difference 𝑑 is equal to two, the coefficient of 𝑛. If 𝑎 sub 𝑛 is equal to two 𝑛 plus four, our sequence is six, eight, 10, and so on. We could then use the formula that 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑. Substituting in our values here gives us 𝑆 sub 10 is equal to 10 over two multiplied by two multiplied by six plus 10 minus one multiplied by two. This simplifies to five multiplied by 12 plus 18, which once again is equal to five multiplied by 30, which gives us an answer of 150.

We will now look at one final question.

Find, in terms of 𝑛, the sum of the arithmetic sequence nine, 10, 11, and so on up to 𝑛 plus eight.

There are two formulas that we can use to calculate the sum of an arithmetic sequence. Firstly, 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by 𝑎 plus 𝑙, where 𝑎 is the first term and 𝑙 is the last term of the sequence. Secondly, 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑. Once again, 𝑎 is the first term and 𝑑 is the common difference of the sequence. We can see that the first term 𝑎 is equal to nine and the last term 𝑙 is 𝑛 plus eight. 𝑆 sub 𝑛 is therefore equal to 𝑛 over two multiplied by nine plus 𝑛 plus eight. Collecting like terms inside the parentheses gives us 𝑛 over two multiplied by 𝑛 plus 17. This is the expression, in terms of 𝑛, for the sum of the arithmetic sequence.

If we chose to use the other formula, we can see from the sequence that the common difference is equal to one. Substituting in our values of 𝑎 and 𝑑 gives us 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by two multiplied by nine plus 𝑛 minus one multiplied by one. The expression inside the brackets simplifies to 18 plus 𝑛 minus one. 18 minus one is equal to 17. Therefore, the expression, once again, is 𝑛 over two multiplied by 𝑛 plus 17. This is the sum of the arithmetic sequence nine, 10, 11, and so on all the way up to 𝑛 plus eight.

We will now summarize the key points from this video. We saw in this video that an arithmetic sequence has first term 𝑎, last term 𝑙, and common difference 𝑑. The 𝑛th term of an arithmetic sequence 𝑎 sub 𝑛 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑. When dealing with a finite arithmetic sequence, this will also be equal to the last term 𝑙. The sum of the terms of an arithmetic sequence is called an arithmetic series. We can calculate this sum denoted 𝑆 sub 𝑛 using one of two formulae, either 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑 or 𝑛 over two multiplied by 𝑎 plus 𝑙. The formula that we choose will depend on the information given in a particular question.