### Video Transcript

In this video, we will learn how to
solve problems involving real-world applications of arithmetic sequences. Weβll see how to find the common
difference, the explicit formula for the πth term, and the order or value of a
specific term in an arithmetic sequence.

Letβs begin by recalling what an
arithmetic sequence is. Well, itβs an ordered list of terms
in which the difference between consecutive terms is constant. For example, the four times table,
four, 8, 12, 16, and so on, is an example of an arithmetic sequence because the
difference between each pair of consecutive terms is four. We call this difference the common
difference, and we represent it using the letter π. We often use the letter π to
represent the first term in the sequence and we have a formula for calculating the
general or πth term π sub π, the πth term, is equal to π plus π minus one
multiplied by π.

This tells us that to calculate any
subsequent term in the sequence, we can take the first term and add on π minus one
lots of the common difference π, which makes sense if we think about it. To find the second term, we need to
add the common difference on once. To find the third term, we have to
add the common difference on twice, so weβre always adding on one less lot of the
common difference than the term number. We also have a formula for
calculating the sum of the first π terms in an arithmetic sequence. π sub π is equal to π over two
multiplied by two π plus π minus one π where, again, π represents the first term
in the sequence and π represents the common difference.

So, those are the basics of
arithmetic sequences which we should already be familiar with. Letβs now look at how we can apply
these results to some real-world problems. In our first example, weβll see how
we can find a specific term in an arithmetic sequence which is presented to us as a
word problem.

Masonβs exercise plan lasts for six
minutes on the first day and increases by four minutes each day. For how long will Mason exercise on
the 18th day?

We can see that Mason increases his
exercise plan by the same amount of four minutes each day. This means that the times Mason
spends exercising daily form an arithmetic sequence with a common difference of
four. Weβre also told that Mason spends
six minutes exercising on the first day of his plan, which means that the first term
of this arithmetic sequence π is equal to six. We therefore have all the
information we need to write down as many terms of the sequence as we want or write
down the rule for the πth term.

The first term in this sequence is
six. The second term is four more than
this, so itβs 10. The third term is four more than
this, so itβs 14. We could continue in this way, but
itβs not very efficient if we need to get all the way to the 18th term in this
sequence. Instead, we can use the formula for
the πth term: π sub π is equal to π plus π minus one multiplied by π. Substituting six for π, the first
term, and four for π, the common difference, we have π sub π is equal to six plus
four multiplied by π minus one.

We could simplify this
algebraically, or to find the 18th term, we could go straight to substituting π
equals 18. π sub 18 is equal to six plus four
multiplied by 18 minus one. We have six plus four multiplied by
17. Four multiplied by 17 is 68. And adding six gives 74. Remember that the terms in the
sequence are times in minutes. So we found that the 18th term of
this sequence or the time Mason spends exercising on the 18th day is 74 minutes.

So in this example, weβve seen how
to calculate a particular term in an arithmetic sequence. In our next example, weβll see how
we can work out the order or term number of a particular term, again from a worded
problem.

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?

Letβs look carefully at the
information weβve been given. Weβre 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 π of
0.5. We donβt 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 sub π equals π plus π minus one
π to form an equation. We have four is equal to π plus
0.5 multiplied by four minus one. That simplifies to four is equal to
π plus 1.5. And we can solve this equation for
π by subtracting 1.5 from each side.

Doing so, we have π is equal to
2.5. So we now know that Olivia ran 2.5
kilometers on the first day of her training. What weβre asked, though, is on
what day will she complete 10 kilometers? So which term in the sequence or
what value of π gives a term equal to 10? We can therefore substitute π
equals 2.5, π equals 0.5, and π π equals 10 to give us an equation we can solve
to find the value of π. Distributing the parentheses, we
have 2.5 plus 0.5π minus 0.5 equals 10. And then the left-hand side
simplifies to two plus 0.5π is equal to 10. We can subtract two from each side
to give 0.5π is equal to eight and then multiply each side of our equation by two
to give π 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 weβd calculated the value of π wouldβve 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 itβs more efficient to use the first method. In either case so, our answer to
the problem is 16.

In our next example, weβll see how
to calculate a specific term in an arithmetic sequence when weβve been given some
information about two of the other terms.

Ameliaβs annual salary increases by
the same quantity every year. In her fourth year at her job, she
earned 24,000 dollars. In her 10th year, she earned 36,000
dollars. How much will she earn in her 20th
year?

If Ameliaβs annual salary increases
by the same amount every year, then her annual salaries form an arithmetic
sequence. We can therefore express the
general term in this sequence using the rule π sub π is equal to π plus π minus
one π where π represents the first term in the sequence, so Ameliaβs salary in her
first year, and π represents the common difference. Thatβs the annual increase. We donβt know either of these
values, but instead weβve been given some information about Ameliaβs salary in the
fourth and 10th years. We can use this information to form
some equations. In the fourth year, she earned
24,000 dollars. So we have the equation 24,000
equals π plus four minus one π or more simply π plus three π.

Weβre also told that in the 10th
year, she earned 36,000 dollars. So we also have the equation 36,000
equals π plus 10 minus one π or π plus 9π. What we now have is a pair of
linear simultaneous equations with two unknowns, π and π. And so we can solve these two
equations simultaneously. By subtracting equation one from
equation two, the π-terms will cancel, and weβre left with 12,000 is equal to
6π. Dividing through by six, and we
found the common difference for this sequence. π is equal to 2,000. So thatβs Ameliaβs annual pay
increase.

To find the value of π, we can sub
π equals 2,000 into either of the two equations. Iβve chosen equation one, giving
24,000 equals π plus three multiplied by 2,000. Subtracting 6,000, thatβs three
multiplied by 2,000, from each side, and we have the value of π. π is equal to 18,000. So that was Ameliaβs salary in the
first year of her job. What weβre asked to find, though,
is what Amelia will earn in her 20th year. So we need to find the 20th term of
this sequence. We can do this by substituting the
values of π and π and the value π equals 20 into our general term formula. We have π sub 20 is equal to
18,000 plus 19. Thatβs 20 minus one multiplied by
2,000. Thatβs 18,000 plus 38,000, which is
56,000. We found then in the 20th year of
her job, Amelia will earn 56,000 dollars assuming she continues to get the same pay
increase of 2,000 dollars every year.

So weβve seen some examples of how
we can calculate a specific term or the order of a term in an arithmetic
sequence. In our next example, weβll practice
finding the sum of the terms in an arithmetic sequence which has been presented as a
word problem.

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 π is the distance run on the first day. Thatβs 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 π terms in an arithmetic sequence can be found using the formula π sub π is
equal to π over two multiplied by two π plus π minus one π. We can therefore substitute 14 for
π, six for π, and 0.5 for π, giving π 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 π terms in an arithmetic sequence, we found the total distance
covered by this runner in 14 days is 129.5 kilometers.

In our final example, weβll see how
we can find the πth term rule for an arithmetic sequence which once again will be
presented as a word problem.

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 π, the πth term
of the sequence which represents the number of minutes that Emma spends exercising
each day. Assume that π equals one is the
first day of Emmaβs plan.

Weβre 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. Weβre 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 π
the πth term of this sequence, so we need to recall the general formula for the
πth term of an arithmetic sequence. Itβs this: π sub π, the πth
term, is equal to π plus π minus one π, where π represents the first term and π
represents the common difference.

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

Letβs now review some of the key
points that weβve covered in this video. Firstly, we reminded ourselves that
in an arithmetic sequence, the difference between consecutive terms is constant and
we call this the common difference and represent it using the letter π. The first term of an arithmetic
sequence is usually denoted by the letter π although it can also be denoted as π
subscript one. We then denote subsequent terms in
the same way: π sub two, π sub three, and so on. We have a formula for calculating
the general or πth term in an arithmetic sequence using the first term, the common
difference, and the term number. π sub π is equal to π plus π
minus one multiplied by π.

We also have a formula for
calculating the sum of the first π terms in an arithmetic sequence: π sub π is
equal to π over two multiplied by two π plus π minus one π. In this video, we saw specifically
how we can apply these results to word problems. We must ensure that we read the
question carefully, identify key information, and then form any equations. We can then solve these equations
to find the general term, a specific term, the order of a term, or the sum of the
first π terms in any arithmetic sequence.