### Video Transcript

Find the sum of the arithmetic series 13 plus 19 plus 25 plus ... plus 85.

The sum of any arithmetic series can be calculated using the formula 𝑆 of 𝑛 is equal to 𝑛 divided by two multiplied by 𝑎 plus 𝑙, where 𝑎 is the first term, 𝑙 is the last term, and 𝑛 is the number of terms in the series. Any individual term 𝑈 𝑛 can be calculated using the formula 𝑎 plus 𝑛 minus one multiplied by 𝑑. 𝑑 in this case stands for the common difference.

In our arithmetic series, the first term 𝑎 is equal to 13, the last term 𝑙 is equal to 85, the common difference of our series is six. As the difference between the first and second terms is six, 13 plus six equals 19. Likewise, 19 plus six equals 25. To get from the second to the third term, we need to add six.

We now need to calculate how many terms are in our series. Well, we know that the last term or the 𝑛th term is equal to 85. Therefore, 𝑎 plus 𝑛 minus one multiplied by 𝑑 is equal to 85. Substituting in our values of 𝑎 and 𝑑 gives us 13 plus six multiplied by 𝑛 minus one equals 85. Subtracting 13 from both sides of this equation leaves us with six multiplied by 𝑛 minus one equals 72.

Dividing both sides of this equation by six gives us 𝑛 minus one equals 12. And finally, adding one to both sides of this equation gives us a value for 𝑛 equal to 13. This means that there 13 terms in our arithmetic series.

We now need to calculate the sum of these 13 terms. Substituting in our values of 𝑛, 𝑎, and 𝑙 gives us 13 divided by two multiplied by 13 plus 85. 13 divided by two is 6.5 and 13 plus 85 is equal to 98. Multiplying 6.5 by 98 gives us an answer of 637.

The sum of the arithmetic series starting at 13 ending in 85 with a common difference of six is 637.