Question Video: Word Problems on Arithmetic Sequences

Ameliaโ€™s annual salary increases by the same quantity every year. In her 4th year at her job, she earned $24,000. In her 10th year, she earned $36,000. How much will she earn in her 20th year?

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Video Transcript

Ameliaโ€™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 Ameliaโ€™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 ๐‘Ž sub ๐‘› is equal to ๐‘Ž plus ๐‘› minus one ๐‘‘ where ๐‘Ž represents the first term in the sequence, so Ameliaโ€™s salary in her first year, and ๐‘‘ represents the common difference. Thatโ€™s the annual increase. We donโ€™t know either of these values, but instead weโ€™ve been given some information about Ameliaโ€™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 ๐‘Ž plus four minus one ๐‘‘ or more simply ๐‘Ž plus three ๐‘‘.

Weโ€™re also told that in the 10th year, she earned 36,000 dollars. So we also have the equation 36,000 equals ๐‘Ž plus 10 minus one ๐‘‘ or ๐‘Ž plus 9๐‘‘. What we now have is a pair of linear simultaneous equations with two unknowns, ๐‘Ž and ๐‘‘. And so we can solve these two equations simultaneously. By subtracting equation one from equation two, the ๐‘Ž-terms will cancel, and weโ€™re left with 12,000 is equal to 6๐‘‘. Dividing through by six, and we found the common difference for this sequence. ๐‘‘ is equal to 2,000. So thatโ€™s Ameliaโ€™s annual pay increase.

To find the value of ๐‘Ž, we can sub ๐‘‘ equals 2,000 into either of the two equations. Iโ€™ve chosen equation one, giving 24,000 equals ๐‘Ž plus three multiplied by 2,000. Subtracting 6,000, thatโ€™s three multiplied by 2,000, from each side, and we have the value of ๐‘Ž. ๐‘Ž is equal to 18,000. So that was Ameliaโ€™s salary in the first year of her job. What weโ€™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 ๐‘Ž and ๐‘‘ and the value ๐‘› equals 20 into our general term formula. We have ๐‘Ž sub 20 is equal to 18,000 plus 19. Thatโ€™s 20 minus one multiplied by 2,000. Thatโ€™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.

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