Video: Properties of Arithmetic Sequences

Olivia is training for a 10 km race. On each training day, she runs 0.5 km more than the previous day. If she completes 4 km on her fourth day, on what day will she complete 10 km?

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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 π‘Ž sub 𝑛 equals π‘Ž plus 𝑛 minus one 𝑑 to form an equation. We have four is equal to π‘Ž plus 0.5 multiplied by four minus one. That simplifies to four is equal to π‘Ž plus 1.5. And we can solve this equation for π‘Ž by subtracting 1.5 from each side.

Doing so, we have π‘Ž 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 π‘Ž equals 2.5, 𝑑 equals 0.5, and π‘Ž 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 π‘Ž 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.

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