- AN EMPLOYEE HAS A SALARY OF $35,500 PER YEAR.
THE EMPLOYEE IS PROMISED
A $2,750 RAISE EACH SUBSEQUENT YEAR.
FIND THE TOTAL EARNINGS OVER AN 8-YEAR PERIOD.
SO THE FIRST THING TO RECOGNIZE ABOUT THIS QUESTION
IS WE'RE TRYING TO FIND THE TOTAL EARNINGS
OVER AN 8-YEAR PERIOD
NOT JUST FIND THE SALARY IN THE EIGHTH YEAR,
AND THEREFORE WE'RE FINDING A PARTIAL SUM
OF A SERIES, NOT A TERM IN A SEQUENCE.
AND BECAUSE THE SALARY IS INCREASING
BY $2,750 EACH YEAR,
WE'RE FINDING A PARTIAL SUM OF AN ARITHMETIC SERIES,
AND THEREFORE WE'LL BE USING THE FORMULAS GIVEN HERE BELOW,
WHERE S SUB N IS THE PARTIAL SUM OF THE FIRST N TERMS
IN AN ARITHMETIC SERIES, D IS THE COMMON DIFFERENCE,
AND "A" SUB N WOULD BE THE Nth TERM IN THE SEQUENCE OR SERIES.
BECAUSE THE SALARY OF THE FIRST YEAR IS 35,500,
"A" SUB 1 = 35,500.
THE SALARY IS INCREASING BY $2,750 EACH YEAR,
AND THEREFORE THE COMMON DIFFERENCE D = 2,750.
AND THIS IS POSITIVE,
BECAUSE THE RAISE IS INCREASING THE SALARY EACH YEAR.
OUR GOAL IS TO FIND THE TOTAL EARNINGS OVER AN 8-YEAR PERIOD,
SO WE'RE TRYING TO FIND S SUB 8,
OR THE SUM OF THE FIRST 8 TERMS IN THIS ARITHMETIC SERIES.
AND BECAUSE WE'RE LOOKING FOR S SUB 8,
THIS ALSO TELLS US N = 8.
NOW IF WE LOOK AT OUR FORMULAS AGAIN,
NOTICE HOW THE PARTIAL SUM FORMULA
REQUIRES US TO HAVE N, WHICH WE HAVE,
"A" SUB 1, WHICH WE HAVE,
BUT WE DON'T HAVE "A" SUB N,
WHICH IN THIS CASE, WOULD BE "A" SUB 8.
WE'LL BEGIN BY DETERMINING "A" SUB 8
AND THEN FIND THE PARTIAL SUM, S SUB 8.
NOTICE HOW THERE ARE TWO FORMULAS FOR "A" SUB N,
BUT THIS LAST FORMULA REQUIRES US TO HAVE "A" SUB N - 1,
OR IN THIS CASE "A" SUB 7, WHICH WE DON'T HAVE.
SO WE'LL USE THIS FORMULA HERE TO FIND "A" SUB 8.
"A" SUB 8 = "A" SUB 1,
35,500 + THE QUANTITY N - 1 = 8 - 1
x THE COMMON DIFFERENCE D, WHICH IS 2,750,
SO WE'D HAVE 35,500 + 7 x 2,750.
SO 35,500 + 7 x 2,750
GETS US "A" SUB 8, WHICH EQUALS 54,750.
THIS WOULD BE THE SALARY DURING THE 8th YEAR.
AND NOW, WE HAVE ALL THE INFORMATION WE NEED
TO FIND THE TOTAL EARNINGS
OVER AN 8-YEAR PERIOD, OR S SUB 8.
THIS IS EQUAL TO N DIVIDED BY 2,
WHICH IN OUR CASE WOULD BE 8 DIVIDED BY 2
x THE QUANTITY "A" SUB 1 + "A" SUB N = 35,500 + 54,750.
LET'S GO BACK TO THE CALCULATOR.
NOTICE HERE, WE HAVE 8 DIVIDED BY 2 = 4.
SO WE HAVE 4 x THE QUANTITY 35,500 + 54,750 = 361,000.
SO IF S SUB 8 = 361,000,
THAT MEANS THIS EMPLOYEE EARNED $361,000 OVER AN 8-YEAR PERIOD.
I HOPE YOU FOUND THIS HELPFUL.