Video Transcript
Olivia is training for a
10-kilometer race. On each training day, she runs 0.5
kilometers more than the previous day. If she completes four kilometers on
her fourth day, on what day will she complete 10 kilometers?
Let’s look carefully at the
information we’ve been given. We’re told that on each training
day, Olivia runs 0.5 or half a kilometer more than the previous day. This means that the distances
Olivia runs each day form an arithmetic sequence with a common difference 𝑑 of
0.5. We don’t know how far Olivia ran on
the first day. But we do know that she runs four
kilometers on the fourth day. We can therefore use the formula
for the general term of an arithmetic sequence 𝑎 sub 𝑛 equals 𝑎 plus 𝑛 minus one
𝑑 to form an equation. We have four is equal to 𝑎 plus
0.5 multiplied by four minus one. That simplifies to four is equal to
𝑎 plus 1.5. And we can solve this equation for
𝑎 by subtracting 1.5 from each side.
Doing so, we have 𝑎 is equal to
2.5. So we now know that Olivia ran 2.5
kilometers on the first day of her training. What we’re asked, though, is on
what day will she complete 10 kilometers? So which term in the sequence or
what value of 𝑛 gives a term equal to 10? We can therefore substitute 𝑎
equals 2.5, 𝑑 equals 0.5, and 𝑎 𝑛 equals 10 to give us an equation we can solve
to find the value of 𝑛. Distributing the parentheses, we
have 2.5 plus 0.5𝑛 minus 0.5 equals 10. And then the left-hand side
simplifies to two plus 0.5𝑛 is equal to 10. We can subtract two from each side
to give 0.5𝑛 is equal to eight and then multiply each side of our equation by two
to give 𝑛 is equal to 16. So the term number or the order of
the term which is equal to 10 is 16. And so we know that Olivia will
complete 10 kilometers on the 16th day of her training plan.
Of course, the other way to answer
this question once we’d calculated the value of 𝑎 would’ve been to list out all the
terms of the sequence by adding 0.5 each time: 2.5, three, 3.5, four. But it would take quite a long time
to get to the 16th term, so it’s more efficient to use the first method. In either case so, our answer to
the problem is 16.
