### Video Transcript

Amelia’s 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 Amelia’s 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 𝑎 sub 𝑛 is equal to 𝑎 plus 𝑛 minus
one 𝑑 where 𝑎 represents the first term in the sequence, so Amelia’s salary in her
first year, and 𝑑 represents the common difference. That’s the annual increase. We don’t know either of these
values, but instead we’ve been given some information about Amelia’s 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 𝑎 plus four minus one 𝑑 or more simply 𝑎 plus three 𝑑.

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

To find the value of 𝑎, we can sub
𝑑 equals 2,000 into either of the two equations. I’ve chosen equation one, giving
24,000 equals 𝑎 plus three multiplied by 2,000. Subtracting 6,000, that’s three
multiplied by 2,000, from each side, and we have the value of 𝑎. 𝑎 is equal to 18,000. So that was Amelia’s salary in the
first year of her job. What we’re 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 𝑎 and 𝑑 and the value 𝑛 equals 20 into our general term formula. We have 𝑎 sub 20 is equal to
18,000 plus 19. That’s 20 minus one multiplied by
2,000. That’s 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.