Question Video: Finding the Sum of a Given Arithmetic Sequence in Terms of 𝑛 Mathematics • 10th Grade

Video Transcript

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

We know that we can calculate the sum of any arithmetic sequence using one of two formulae. Firstly, 𝑠 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑. In this formula, 𝑎 is the first term and 𝑑 is the common difference. Alternatively, 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.

We can see from our sequence that the first term 𝑎 is nine and the common difference is one as the numbers are increasing by one. The value of our last term 𝑙 is 𝑛 plus eight. Substituting in our values of 𝑎 and 𝑑 to the first formula gives us 𝑛 over two multiplied by two multiplied by nine plus 𝑛 minus one multiplied by one. This simplifies to 𝑛 over two multiplied by 18 plus 𝑛 minus one. We can simplify the square bracket further such that the sum of the first 𝑛 terms is equal to 𝑛 over two multiplied by 𝑛 plus 17.

If we chose to use the second formula, we need to substitute 𝑎 equals nine and 𝑙 equals 𝑛 plus eight. This gives us 𝑠 sub 𝑛 is equal to 𝑛 over two multiplied by nine plus 𝑛 plus eight. Once again, the bracket or parentheses simplifies to 𝑛 plus 17.

The sum of the arithmetic sequence in terms of 𝑛 is 𝑛 over two multiplied by 𝑛 plus 17.