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: 𝑎 sub 𝑛, the 𝑛th
term, is equal to 𝑎 plus 𝑛 minus one 𝑑, where 𝑎 represents the first term and 𝑑
represents the common difference.
We can therefore substitute the
values of 𝑎 and 𝑑, which we were given in the question to find our general
term. It’s 𝑎 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.
