# Question Video: Finding the Sum of an Arithmetic Sequence in a Real-World Context

A runner is preparing himself for a long-distance race. He covered 6 km on the first day and then increased the distance by 0.5 kilometers every day. Find the total distance he covered in 14 days.

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### Video Transcript

A runner is preparing himself for a long-distance race. He covered six kilometers on the first day and then increased the distance by 0.5 kilometers every day. Find the total distance he covered in 14 days.

As the distance run increases by the same amount every day, these distances form an arithmetic sequence. The common difference for this sequence is 0.5, and the first term 𝑎 is the distance run on the first day. That’s six kilometers. To find the total distance covered in 14 days, we need to find the sum of the first 14 terms in this sequence. We recall then that the sum of the first 𝑛 terms in an arithmetic sequence can be found using the formula 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one 𝑑. We can therefore substitute 14 for 𝑛, six for 𝑎, and 0.5 for 𝑑, giving 𝑆 sub 14 is equal to 14 over two multiplied by two times six plus 0.5 multiplied by 14 minus one.

That simplifies to seven multiplied by 12 plus 0.5 multiplied by 13. We keep going inside the parentheses. We have 12 plus 6.5, which is 18.5, and then multiplying by seven gives 129.5. Remember, this is a distance and the units are kilometers. So by applying the formula for the sum of the first 𝑛 terms in an arithmetic sequence, we found the total distance covered by this runner in 14 days is 129.5 kilometers.