Asked by Emily
In a geometric series, t1=12 and S3=372. What is the greatest possible value for t5? Justify your answer.
Answers
Answered by
Reiny
so, we know
a = 12 and
372 = 12(r^3 - 1)/(r-1)
31r - 31 = r^n - 1
r^3 - 31r+ 30 = 0
by inspection I saw that r=1 works
so r-1 is a factor
by synthetic division
I got (r-1)(r-5)(r+6) = 0
r = 1, 5, -6
if r=1 , t5 = ar^4 = 12(1^4) = 12 , all terms would stay the same
if r=5, t5= 12(5)^4 = 7500
if r = -6 , t5 = 12(-6)^4 = + 15552 -- the greatest
btw, if r = -6
terms are : 12 , -72 , 432 , -2592 ...
and 12 - 72 + 432 = 372
a = 12 and
372 = 12(r^3 - 1)/(r-1)
31r - 31 = r^n - 1
r^3 - 31r+ 30 = 0
by inspection I saw that r=1 works
so r-1 is a factor
by synthetic division
I got (r-1)(r-5)(r+6) = 0
r = 1, 5, -6
if r=1 , t5 = ar^4 = 12(1^4) = 12 , all terms would stay the same
if r=5, t5= 12(5)^4 = 7500
if r = -6 , t5 = 12(-6)^4 = + 15552 -- the greatest
btw, if r = -6
terms are : 12 , -72 , 432 , -2592 ...
and 12 - 72 + 432 = 372
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