- WELCOME TO A LESSON ARITHMETIC SEQUENCES.
THE GOALS OF THIS VIDEO ARE TO DETERMINE THE Nth TERM
OF AN ARITHMETIC SEQUENCE,
DETERMINE THE COMMON DIFFERENCE,
AND ALSO TO DETERMINE THE FORMULA
OF AN ARITHMETIC SEQUENCE.
AS WE LEARNED IN THE PREVIOUS VIDEO,
AN ARITHMETIC SEQUENCE IS A SEQUENCE
THAT HAS A PATTERN OF ADDING A CONSTANT
TO DETERMINE CONSECUTIVE TERMS.
WE CAN SAY THAT ARITHMETIC SEQUENCES
HAVE A COMMON DIFFERENCE.
SO IF WE TAKE A LOOK AT THIS SEQUENCE DOWN HERE
AND LOOK FOR A PATTERN,
WHAT WE'LL NOTICE IS THAT THERE'S A PATTERN
OF ADDING FOUR TO OBTAIN THE NEXT TERM IN THE SEQUENCE.
AND SO BECAUSE THE PATTERN STAYS CONSISTENT OF ADDING A CONSTANT,
THIS IS AN ARITHMETIC SEQUENCE.
WHAT WE'LL NOTICE IS WE CAN FIND THE COMMON DIFFERENCE
FROM THIS FORMULA BY SELECTING ANY TERM IN THE SEQUENCE
AND SUBTRACTING THE TERM BEFORE IT.
SO IF WE USE "A" SUB 2, WE WOULD SUBTRACT "A" SUB 1.
-1 - -5 WILL GIVE US 4.
IF WE WANT TO USE THE FIFTH TERM IN THE SEQUENCE,
WE WOULD HAVE TO SUBTRACT THE FOURTH TERM.
11 - 7 ALSO EQUALS 4.
THIS NEXT FORMULA, "A" SUB N = "A" SUB 1 + (N - 1)
TIMES THE COMMON DIFFERENCE,
YOU MIGHT BE THINKING, WHY THE N -1?
WELL, IF YOU TAKE A LOOK AT THE SEQUENCE AGAIN,
LOOK AT THE PATTERN, NOTICE THE SECOND TERM,
WE ADDED FOUR ONE TIME.
THE THIRD TERM, WE WOULD ADD FOUR TWO TIMES.
AND THE FOURTH TERM, WE WOULD ADD FOUR THREE TIMES.
SO WE'LL ALWAYS ADD
N - 1 MULTIPLES OF THE COMMON DIFFERENCE TO THE FIRST TERM
TO GET ANY OTHER TERM IN THE SEQUENCE.
AND THIS LAST FORMULA HERE IS A RECURSIVE FORMULA
WHICH JUST MEANS IF THEY WANT TO FIND, LET'S SAY,
THE FIFTH TERM, WE JUST TAKE THE FOURTH TERM, "A" SUB N - 1,
AND THEN ADD THE COMMON DIFFERENCE.
AND THIS IS JUST EMPHASIZING THE PATTERN THAT WE SEE HERE IN RED.
LET'S GO AHEAD AND SEE IF WE CAN ANSWER SOME QUESTIONS
USING ARITHMETIC SEQUENCES.
AGAIN, AS WE STATED IN THE PREVIOUS VIDEO,
A SEQUENCE IS A FUNCTION,
SO WE CAN IDENTIFY THE DOMAIN AND THE RANGE.
AND WHAT'S HELPFUL TO DETERMINE THE DOMAIN AND RANGE
IS TO LABEL EACH TERM AS
"A" SUB 1, "A" SUB 2, "A" SUB 3, "A" SUB 4, "A" SUB 5,
AND, LASTLY, "A" SUB 6.
SO THE VALUES OF N REPRESENT THE DOMAIN,
SO THE DOMAIN OF THIS SEQUENCE WILL BE 1, 2, 3, 4, 5, AND 6.
AND THE RANGE OF THE SEQUENCE WOULD BE THE ACTUAL ELEMENTS
IN THE LIST.
AND WE USUALLY LIST THESE FROM LEAST TO GREATEST,
SO I'LL LIST THE RANGE AS -6, -3, 0, 3, 6, AND 9.
OKAY, ON THIS PROBLEM,
WE'RE GIVEN THE FORMULA FOR THE ARITHMETIC SEQUENCE.
WE WANT TO DETERMINE THE FIRST THREE TERMS
AND THEN THE EIGHTH TERM,
SO WE WANT TO FIND "A" SUB 1, "A" SUB 2,
"A" SUB 3 AND "A" SUB 8.
SO IF WE WANT "A" SUB 1, THAT MEANS N IS EQUAL TO 1.
SO WE HAVE -4 x 1 + 3, THAT'S -1.
"A" SUB 2, THAT MEANS N IS EQUAL TO 2.
-4 x 2 + 3.
-8 + 3 = -5.
"A" SUB 3, SO N IS 3.
WE HAVE -12 + 3. THAT'S -9.
AND THE EIGHT TERM, THAT MEANS N IS EQUAL TO EIGHT.
-4 x 8 + 3.
YOU HAVE -32 + 3, THAT'S -29.
SO THE SEQUENCE WOULD LOOK LIKE THIS, -1, -5, -9,
AND THEN THE EIGHTH TERM WOULD BE -29.
THEY ALSO ASKED US TO STATE THE COMMON DIFFERENCE.
REMEMBER, TO FIND THE COMMON DIFFERENCE,
WE JUST SELECT ONE TERM AND THEN SUBTRACT THE TERM BEFORE IT.
SO IF WE USE "A" SUB 2,
THE COMMON DIFFERENCE WILL BE -5 - "A" SUB 1--IT'S -1--
SO WE HAVE -5 + 1, SO THE COMMON DIFFERENCE IS -4.
AND WE CAN SEE THIS SEQUENCE IS DECREASING BY 4 EACH TIME
BECAUSE WE'RE ADDING -4 EACH TIME.
LET'S GO AND TAKE A LOOK AT THIS QUESTION NOW.
WE'RE GIVEN THAT IT'S AN ARITHMETIC SEQUENCE.
WE WANT TO DETERMINE THE FORMULA FOR THE SEQUENCE
AND THEN ALSO DETERMINE THE 12th TERM.
TO DETERMINE THE FORMULA FOR THE SEQUENCE,
WE'RE GOING TO HAVE TO DETERMINE THE FORMULA FOR "A" SUB N
AS WE SEE HERE.
SO WE'RE GOING TO HAVE TO FIND "A" SUB 1
AND ALSO THE COMMON DIFFERENCE TO USE THIS FORMULA.
SO LET'S GO AHEAD AND WRITE THIS OUT.
"A" SUB N = "A" SUB 1 + (N - 1) x D.
"A" SUB 1 IS THE FIRST TERM, SO "A" SUB 1 = -2.
AND TO DETERMINE THE COMMON DIFFERENCE,
WE'RE GOING TO SELECT ANY TERM
AND THEN SUBTRACT THE TERM BEFORE IT.
SO IF I WANTED TO USE THE TERM HERE THAT'S EQUAL TO 12,
THE COMMON DIFFERENCE WOULD BE
12 MINUS THE TERM BEFORE IT OR 8.5.
SO THE COMMON DIFFERENCE IS EQUAL TO 3.5.
REMEMBER, I CAN OBTAIN THE COMMON DIFFERENCE
BY SELECTING ANY TERM AND SUBTRACTING THE TERM BEFORE IT.
SO IF I WANTED TO USE THIS TERM HERE,
1.5 - -2 IS STILL GOING TO EQUAL 3.5.
NOW THAT WE HAVE "A" SUB 1 AND D,
WE CAN SUBSTITUTE THOSE VALUES INTO THE FORMULA
AND FIND "A" SUB N.
SO "A" SUB N = -2 + N - 1 TIMES A COMMON DIFFERENCE OF 3.5.
NOW THIS IS THE FORMULA
TO DETERMINE ANY TERM IN THE SEQUENCE.
BUT NORMALLY, WE'LL BE ASKED TO SIMPLIFY THIS,
SO WE'LL DISTRIBUTE 3.5 AND THEN COMBINE ANY LIKE TERMS.
SO WE'LL HAVE "A" SUB N = -2 + 3.5N - 3.5.
HERE WE HAVE TWO LIKE TERMS,
SO OUR FORMULA WILL BE "A" SUB N = 3.5N - 5.5.
ONE MORE THING TO NOTICE HERE.
YOU TAKE A LOOK AT THE COEFFICIENT OF THE N TERM.
IT IS EQUAL TO THE COMMON DIFFERENCE.
WE WERE ALSO ASKED TO DETERMINE THE 12th TERM.
SO USING THIS FORMULA, WE'LL LET N = 12 TO FIND THAT 12th TERM.
LET'S GO AND DO THAT ON THE NEXT SCREEN.
SO HERE'S THE FORMULA THAT WE HAD ON THE PREVIOUS SCREEN,
SO "A" SUB 12 = 3.5 x 12 - 5.5.
WELL, 3.5 x 12, THAT'LL BE 42 - 5.5,
WHICH EQUALS 36.5.
SO THE 12th TERM IS EQUAL TO 36.5.
OKAY, THAT'LL DO IT FOR THIS VIDEO.
NEXT, WE'LL TAKE A LOOK AT GEOMETRIC SEQUENCES.