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- A POPULATION GROWS ACCORDING TO AN EXPONENTIAL GROWTH MODEL.
THE INITIAL POPULATION IS P SUB 0 = 16
AND THE GROWTH RATE IS R = 0.35.
WHICH MEANS THE GROWTH RATE WOULD BE 35%.
SO EVERY END UNIT OF TIME THE POPULATION GROWS
OR INCREASES BY 35%.
BUT FOR THE RECURSIVE AND EXPLICIT FORMULAS GIVEN HERE,
WE DO NEED TO USE THE DECIMAL FORM OF THE GROWTH RATE.
SO FOR THE RECURSIVE FORMULA
WE WOULD HAVE P SUB N = THE QUANTITY 1 + R
THAT WOULD BE 1 + 0.35 SO WE'D HAVE 1.35 x P SUB AND - 1.
SO NOW WHEN WE FIND P SUB 1, N = 1,
SO I'LL SUBSTITUTE 1 HERE AS WELL AS HERE.
SO P SUB 1 IS = TO 1.35 x P SUB 1 - 1
THAT WOULD BE P SUB O WHICH WE KNOW IS EQUAL TO 16.
SO 1.35 x 16 = 21.6. SO P SUB 1 = 21.6.
NOW WHEN WE FIND P SUB 2, N = 2.
SO WE'LL HAVE 1.35 x P SUB 2 - 1 THAT'S P SUB 1,
WHICH WE JUST FOUND.
THAT'S EQUAL TO 21.6.
AND NOW WE'LL GO BACK TO THE CALCULATOR
1.35 x 21.6 = 29.16.
NOTICE WHEN USING THE RECURSIVE FORMULA
TO FIND THE NEXT VALUE OF P
WE ALWAYS USE THE PREVIOUS VALUE OF P.
NEXT WE'RE ASKED TO FIND THE EXPLICIT FORMULA FOR P SUB N,
WHICH IS GIVEN HERE FOR EXPONENTIAL GROWTH.
P SUB N = P SUB 0 x THE QUANTITY 1 + R RAISE TO THE POWER OF N.
AND WE ALREADY HAVE ALL THE INFORMATION WE NEED.
P SUB 0 = 16 AND R = 0.35.
SO WE HAVE P SUB N = P SUB 0 WHICH IS 16 x THE QUANTITY 1 + R
WHICH IS 1.35 RAISE TO THE POWER OF N.
NOW WE CAN USE OUR EXPLICIT FORMULA TO FIND P SUB 11
WITHOUT KNOWING P SUB 10.
P SUB 11 IS = TO 16 x 1.35 RAISE TO THE POWER OF 11.
SO 16 x 1.35 RAISE TO THE POWER OF 11.
OUR DIRECTIONS DO SAY ROUND TO AT LEAST 1 DECIMAL PLACE.
SO IF ROUND TO THE TENTHS,
NOTICE HOW WE HAVE A 0 IN THE HUNDREDTHS
SO THIS WOULD BE APPROXIMATELY 434.3.
I HOPE YOU FOUND THIS HELPFUL.