Question Video: Solving Word Problems by Finding the General Term of a Sequence Mathematics

Emma started a workout plan to improve her fitness. She exercised for fourteen minutes on the first day and increased the duration of her exercise plan by six minutes each subsequent day. Find, in terms of 𝑛, the 𝑛th term of the sequence which represents the number of minutes that Emma spends exercising each day. Assume that 𝑛 = 1 is the first day of Emma’s plan.

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Video Transcript

Emma started a workout plan to improve her fitness. She exercised for 14 minutes on the first day and increased the duration of her exercise plan by six minutes each subsequent day. Find, in terms of 𝑛, the 𝑛th term of the sequence which represents the number of minutes that Emma spends exercising each day. Assume that 𝑛 equals one is the first day of Emma’s plan.

We’re told in this problem that Emma increased her exercise by the same amount every day, which means that the time spent exercising form an arithmetic sequence with a common difference of six. We’re also told that Emma exercised for 14 minutes on the first day of her plan, which means the first term in the sequence is 14. We are asked to find in terms of 𝑛 the 𝑛th term of this sequence, so we need to recall the general formula for the 𝑛th term of an arithmetic sequence. It’s this: π‘Ž sub 𝑛, the 𝑛th term, is equal to π‘Ž plus 𝑛 minus one 𝑑, where π‘Ž represents the first term and 𝑑 represents the common difference.

We can therefore substitute the values of π‘Ž and 𝑑, which we were given in the question to find our general term. It’s π‘Ž sub 𝑛 is equal to 14 plus six multiplied by 𝑛 minus one. Now it’s usual to go on and simplify algebraically. So we’ll distribute the parentheses. We have 14 plus six 𝑛 minus six, which then simplifies to six 𝑛 plus eight. And it is usual to give the general term of an arithmetic sequence in this form, some multiple of 𝑛 plus a constant. Notice as well that that common difference of six is the coefficient of 𝑛 in our general term and that will always be the case for an arithmetic sequence. We found the 𝑛th term of this sequence. It’s six 𝑛 plus eight. And by substituting any value of 𝑛, we can calculate any term in this sequence.

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