- WELCOME TO A VIDEO ON ARITHMETIC SERIES.
THE GOALS OF THIS VIDEO ARE TO DEFINE A SERIES,
AND ALSO TO DETERMINE THE PARTIAL SUM
OF AN ARITHMETIC SERIES.
THE SUMMING OR ADDING OF THE TERMS
IN AN ARITHMETIC SEQUENCE CREATES
WHAT IS CALLED A SERIES.
SO A SEQUENCE IS A LIST OF TERMS,
AND A SERIES IS A SUM OF TERMS.
AND THESE ARE THE FORMULAS THAT WE CAN USE
TO FIND THE Nth PARTIAL SUM OF AN ARITHMETIC SEQUENCE.
LET'S SEE IF WE CAN FIGURE OUT
WHERE THESE FORMULAS ARE COMING FROM,
AND LET'S DO THAT BY CONSIDERING THIS PROBLEM.
LET'S SAY WE WANT TO FIND THIS SUM,
BUT WE WANT TO FIND AN EASIER WAY TO DO IT
THEN JUST ADD ALL OF THESE NUMBERS.
WELL, THE FIRST THING YOU MIGHT NOTICE
IS THAT THE SERIES IS AN ARITHMETIC SERIES,
BECAUSE EACH TERM INCREASES BY 1.
SO OUR COMMON DIFFERENCE IS + 1.
THE NEXT THING WE MIGHT DO
IS START PAIRING UP THESE NUMBERS.
IF WE START PAIRING THESE UP,
WE'RE GOING TO NOTICE A PATTERN.
WHAT I MEAN BY THAT IS IF WE TAKE THE FIRST TERM
AND ADD IT TO THE LAST TERM, THAT HAS A SUM OF 101.
AND IF WE TAKE THE SECOND TERM
AND ADD IT TO THE SECOND TO THE LAST TERM,
WE'LL ALSO HAVE A SUM OF 101, AND THIS PATTERN CONTINUES,
SO WE COULD CONTINUE PAIRING THESE UP TO HAVE SUMS OF 101.
THEN THE QUESTION BECOMES HOW MANY 101s WOULD WE HAVE?
WELL, IF THERE ARE 100 TERMS, AND WE'RE MAKING GROUPS OF 2,
THAT WOULD LEAVE US WITH 50 GROUPS OF 101.
SO THE SUM OF ALL OF THESE TERMS WOULD BE 50 x 101,
WHICH = 5,050.
NOW, IF WE LOOK AT THIS IN THE MOST GENERAL SENSE,
MEANING WE WANT THE SUM OF N NUMBER OF TERMS,
TO FIND THE SUM WE'RE GOING TO CREATE
BY PAIRING THE FIRST AND LAST TERM
WOULD BE "A" SUB 1 + "A" SUB N.
AND THE NUMBER OF PAIRS WE WOULD HAVE
WOULD BE THE TOTAL NUMBER OF TERMS,
WHICH IS N DIVIDED BY 2.
AND THIS IS ONE OF THOSE FORMULAS
THAT WE CAN USE TO FIND THE PARTIAL SUM
OF AN ARITHMETIC SERIES.
NOW, LET'S TAKE A LOOK AT THE SECOND FORMULA.
ONE THING WE'LL NOTICE IS THAT THIS FORMULA REQUIRES
THE LAST TERM IN THE SERIES, AND THIS FORMULA DOESN'T
BUT, OF COURSE, THEY ARE RELATED.
IF WE WERE TO TAKE A CLOSER LOOK AT THIS FORMULA,
WE COULD REWRITE 2 x "A" SUB 1 AS "A" SUB 1 + "A" SUB 1,
LEAVE EVERYTHING ELSE THE SAME.
AND IF YOU REMEMBER FROM THE VIDEO ON ARITHMETIC SEQUENCES,
"A" SUB 1 + "A" SUB N - 1 x D = "A" SUB N.
SO THIS WOULD JUST BECOME S SUB N = N x "A" SUB 1
+ "A" SUB N/2,
WHICH COULD BE REWRITTEN AS THIS FIRST FORMULA HERE.
LET'S GO AHEAD AND APPLY THESE FORMULAS.
LET'S SAY WE WANT TO FIND THE SUM OF THIS ARITHMETIC SERIES.
LET'S WRITE DOWN ALL THE INFORMATION THAT WE KNOW.
"A" SUB 1 IS THE FIRST TERM, THAT'S EQUAL TO 3.
THE LAST TERM WE'LL CALL" A" SUB N = 73.
AND WE KNOW IT'S ARITHMETIC,
SO WE CAN FIND THE COMMON DIFFERENCE.
LET'S GO AHEAD AND TAKE 8 AND SUBTRACT 3.
THAT WOULD BE D = 5.
REMEMBER, WE CAN CHOOSE ANY TERM
AND THEN SUBTRACT THE TERM BEFORE IT
TO FIND THE COMMON DIFFERENCE.
NOW, IF WE TAKE A LOOK AT THE FORMULA
TO FIND THE PARTIAL SUM,
WE DO HAVE TO FIND THE VALUE OF N STILL,
BUT WE CAN USE THIS FORMULA HERE TO FIND THE VALUE OF N.
IF WE LET "A" SUB N = 73, WE SHOULD BE ABLE TO SOLVE FOR N.
LET'S GIVE IT A TRY.
"A" SUB 1 = 3.
WE'RE TRYING TO FIND N,
AND THE COMMON DIFFERENCE IS EQUAL TO 5.
WELL, HERE WE HAVE ONE UNKNOWN.
SO WE SHOULD BE ABLE TO SOLVE FOR N.
DISTRIBUTE THE 5, COMBINE LIKE TERMS.
THIS WILL BE 3 - 5.
THAT'LL BE 5N - 2.
ADD 2 TO BOTH SIDES.
IT'LL BE 75.
DIVIDE BY 5.
IT LOOKS LIKE N = 15.
NOW WE HAVE ALL THE INFORMATION WE NEED
TO FIND THIS PARTIAL SUM.
THE SUM OF THE FIRST 15 TERMS WILL = N DIVIDED BY 2,
15 DIVIDED BY 2 x THE QUANTITY OF "A" SUB 1,
WHICH IS 3 + "A" SUB 15, WHICH IS "A" SUB N.
SO THAT'S 73.
AND SO WE'RE GOING TO HAVE 15/2 TIMES--THIS WILL BE 76.
SO I'LL WRITE 76/1.
SIMPLIFY THIS.
THAT'LL BE 1.
THAT'LL BE 38.
SO THE SUM OF THE FIRST 15 TERMS = 15 x 38, WHICH = 570.
LET'S GO AHEAD AND TAKE A LOOK AT ANOTHER EXAMPLE.
HERE WE'RE GIVEN THE FORMULA FOR AN ARITHMETIC SEQUENCE
AND WE WANT TO FIND THE SUM OF THE ARITHMETIC SERIES
THAT IT CREATES.
SO, AGAIN, LOOKING AT OUR PARTIAL SUM FORMULA,
WE WANT THE SUM OF THE FIRST 20 TERMS, SINCE N IS 20
AND THAT'S GOING TO BE EQUAL TO N
DIVIDED BY 2 x "A" SUB 1 + "A" SUB N.
SO LET'S GO AHEAD AND WRITE DOWN ALL THE INFORMATION
THAT WE KNOW.
WE KNOW N IS 20, AND WE DON'T KNOW "A" SUB 1,
BUT WE CAN FIND IT WITH THIS FORMULA.
"A" SUB 1 = -4 x 1 + 3.
SO "A" SUB 1 = -4 + 3.
THAT'S -1.
WE ALSO NEED "A" SUB N, WHICH MEANS WE NEED "A" SUB 20.
SO WE'LL HAVE -4 x 20 + 3.
THAT'LL BE -80 + 3.
THAT'S -77.
SO WE HAVE N, WE HAVE "A" SUB 1, AND WE HAVE "A" SUB 20.
THAT'S ALL WE NEED.
SO THE SUM OF THE FIRST 20 TERMS = 20
DIVIDED BY 2 x (-1 + -77).
SO WE'RE GOING TO HAVE 10 TIMES--THIS WILL BE -78.
SO THE PARTIAL SUM IS EQUAL TO -780.
OKAY. THAT'S GOING TO DO IT FOR THIS VIDEO.
I HOPE YOU FOUND IT HELPFUL,
AND THANK YOU FOR WATCHING.