- IF THE 16th TERM OF A GEOMETRIC SEQUENCE IS 1,024,
AND THE COMMON RATIO IS 2,
WE WANT TO FIND THE SEQUENCE FORMULA
AND THE FIRST THREE TERMS.
SO BECAUSE WE'RE TOLD THIS IS A GEOMETRIC SEQUENCE,
WE KNOW THE SEQUENCE FORMULA
MUST BE IN THE FORM OF A SUB N EQUALS A SUB 1 x R
RAISED TO THE POWER OF N - 1.
SO NOW IF WE GO BACK AND LOOK AT THE GIVEN INFORMATION,
WE'RE TOLD THE 16th TERM OF THE SEQUENCE IS 1,024
WHICH MEANS A SUB 16 MUST EQUAL 1,024.
WE'RE ALSO TOLD THE COMMON RATIO R IS EQUAL TO 2.
AND NOW WE'LL USE THIS INFORMATION
TO PERFORM SUBSTITUTION INTO OUR SEQUENCE FORMULA.
IF WE LET A SUB N BE EQUAL TO A SUB 16,
WE CAN SUBSTITUTE 1,024 FOR A SUB N
AS LONG AS WE SUBSTITUTE 16 FOR N,
SO WE'D HAVE 1,024 EQUALS A SUB 1 x R
WHICH WE KNOW IS 2 RAISED TO THE POWER OF N - 1
WHERE AGAIN, N = 16, SO WE HAVE 16 - 1 AS OUR EXPONENT.
SO WE HAVE 1,024 EQUALS TO THIS WOULD BE
A SUB 1 x 2 TO THE 15th.
SO NOW TO SOLVE FOR A SUB 1
WE'LL DIVIDE BOTH SIDES BY 2 RAISED TO THE POWER OF 15.
WHEN WE DO THIS WE'LL GET THE FIRST TERM,
BUT WE'LL ALSO HAVE ENOUGH INFORMATION
TO WRITE OUR SEQUENCE FORMULA.
THIS WOULD SIMPLIFY TO 1,
SO WE HAVE A SUB 1 EQUALS THIS QUOTIENT HERE.
LET'S GO AHEAD AND USE OUR CALCULATOR.
WE HAVE 1,024 DIVIDED BY 2 RAISED TO THE POWER OF 15.
LET'S GO AHEAD AND CONVERT THIS TO A FRACTION,
SO A SUB 1 IS EQUAL TO 1/32
WHICH MEANS OUR FORMULA FOR THE SEQUENCE MUST BE A SUB N
EQUALS A SUB 1 WHICH WE JUST FOUND TO BE 1/32 x R
WHICH WE KNOW IS 2 RAISED TO THE N - 1 POWER.
NOW REALIZE WE COULD CHANGE THE FORM OF THIS
SINCE 32 IS EQUAL TO 2 TO THE 5th,
BUT LET'S GO AHEAD AND LEAVE IT LIKE THIS.
SO HERE'S THE SEQUENCE FORMULA.
HERE'S THE FIRST TERM,
BUT WE'RE ALSO ASKED TO FIND THE NEXT TWO TERMS.
SO WE COULD USE A SUB 1
AND MULTIPLY BY THE COMMON RATIO
TO FIND THE SECOND AND THIRD TERM,
BUT LET'S GO AHEAD AND PRACTICE USING OUR FORMULA.
SO OUR SECOND TERM A SUB 2
IS GOING TO EQUAL TO 1/32 x 2 TO THE 2 - 1 POWER OR FIRST POWER.
THIS WOULD BE 2/32 OR 1/16.
AND THEN FOR A SUB 3,
OUR THIRD TERM, WE'D HAVE 1/32 x 2 RAISED TO THE POWER OF 3 - 1
OR THE SECOND POWER.
2 SQUARED IS 4, SO WE'D HAVE 4 OVER 32 WHICH SIMPLIFIES TO 1/8.
SO AGAIN, HERE'S THE FORMULA FOR OUR SEQUENCE,
AND THE FIRST THREE TERMS WOULD BE 1/32, 1/16, AND 1/8.
OF COURSE, THE SEQUENCE WOULD CONTINUE,
BUT THIS IS WHAT THE QUESTION ASKS FOR.
NOW THAT WE HAVE THE SEQUENCE AS A LIST,
NOTICE HOW THE PATTERN HERE TO FIND CONSECUTIVE TERMS
IS TO MULTIPLY BY 2, OUR COMMON RATIO.
OKAY. I HOPE YOU FOUND THIS EXPLANATION HELPFUL.