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.