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- A POPULATION GROWS ACCORDING TO AN EXPONENTIAL GROWTH MODEL
WITH P SUB 0 = 70 AND P SUB 1 = 98.
WE WANT TO FIND THE RECURSIVE FORMULA,
THE EXPLICIT FORMULA, AND THEN FIND P OF 5.
BOTH THE RECURSIVE AND EXPLICIT FORMULAS
FOR EXPONENTIAL GROWTH ARE GIVEN HERE.
NOTICE IN BOTH CASES,
WE NEED TO FIND THE QUANTITY 1 + R.
WHICH IS THE GROWTH FACTOR
OR AT LEAST R, THE GROWTH RATE, TO FIND 1 + R.
SO WE HAVE THE OPTION OF USING THE FORMULA HERE
FOR THE GROWTH FACTOR 1 + R
OR WE CAN USE THIS FORMULA HERE TO FIND R AND THEN FIND 1 + R.
WE NORMALLY WOULDN'T FIND BOTH
BUT AS AN EXAMPLE I WILL SHOW BOTH.
LET'S BEGIN BY DETERMINING THE GROWTH FACTOR, 1 + R.
1 + R = P SUB N DIVIDED BY P SUB N - 1.
NOTICE THAT P SUB N - 1 WOULD BE THE TERM BEFORE P SUB N.
SO IF WE USE P SUB 1 FOR P SUB N
WE'D HAVE P SUB 1 DIVIDED P SUB 0.
NOTICE HOW WE HAVE BOTH P SUB 1 AND P SUB 0.
THIS WOULD BE 98 DIVIDED BY 70
AND NOW WE'LL GO TO THE CALCULATOR,
98 DIVIDED BY 70 = 1.4.
SO IF 1 + R = 1.4 WE SHOULD RECOGNIZE THAT R = 0.4
AND THEREFORE THE GROWTH RATE WOULD BE 40%.
LET'S GO AHEAD AND FIND R ANYWAY USING THIS FORMULA HERE.
R WOULD BE EQUAL TO--AGAIN, IF P SUB N = P SUB 1,
WE'D HAVE P SUB 1 - P SUB 0 DIVIDED BY P SUB 0.
AGAIN, IF N IS 1, THIS WOULD BE P SUB 1, THIS IS P SUB 0.
THIS IS ALSO P SUB 0.
SO WE'D HAVE 98 - 70 DIVIDED BY 70,
WHICH WOULD BE 28 DIVIDED BY 70 AND 28 DIVIDED BY 70 = 0.4.
AGAIN, NORMALLY WE WOULD NOT FIND BOTH OF THESE.
WE WOULD PROBABLY JUST FIND THE GROWTH FACTOR OF 1 + R
BECAUSE THAT'S WHAT'S NEEDED IN BOTH FORMULAS.
SO NOW FOR THE RECURSIVE FORMULA
WE'D HAVE P SUB N = OUR GROWTH FACTOR WHICH WE JUST FOUND
AS 1.4 x P SUB N - 1.
AND NOW LET'S FIND THE EXPLICIT FORMULA WHICH IS GIVEN HERE.
WE'D HAVE P SUB N = P SUB 0,
WHICH IS GIVEN AS 70 x THE GROWTH FACTOR
OR THE QUANTITY OF 1 + R RAISE TO THE POWER OF N.
SO WE HAVE 1.4 RAISE TO THE POWER OF N.
SO HERE'S OUR RECURSIVE FORMULA FOR THE EXPONENTIAL GROWTH
AND HERE'S THE EXPLICIT FORMULA FOR EXPONENTIAL GROWTH.
AND NOW TO FIND P SUB 5 WE'LL USE THE EXPLICIT FORMULA HERE
BECAUSE IF I USE THE RECURSIVE FORMULA,
NOTICE HOW WE'D HAVE TO HAVE P SUB 4, WHICH WE DON'T HAVE.
WE COULD FIND IT BUT IT WOULD TAKE QUITE A BIT OF WORK.
SO USING THE EXPLICIT FORMULA
P SUB 5 = 70 x 1.4 RAISE TO THE 5th.
SO AGAIN, WE HAVE 70 x 1.4 RAISED TO THE 5th POWER.
AND OUR DIRECTIONS DO TELL US
TO ROUND TO AT LEAST 1 DECIMAL PLACE.
SO WE ROUND TO THE 10th PLACE VALUE.
NOTICE HOW THE 7 IN THE HUNDREDTHS PLACE VALUE
INDICATIVES TO ROUND UP.
SO WE'LL SAY P SUB 5 IS APPROXIMATELY 376.5.
I HOPE YOU FOUND THIS HELPFUL.