Question Video: Writing an Expression for the Sum of an Arithmetic Sequence Mathematics

Video Transcript

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.