SOLUTION: Correction: An infinite geometric series has the first term u1 = a and u2 = (1/4a^2)-3a, where a > 0.
a. Find the values of a for which the sum to infinity of the series exists.
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-> SOLUTION: Correction: An infinite geometric series has the first term u1 = a and u2 = (1/4a^2)-3a, where a > 0.
a. Find the values of a for which the sum to infinity of the series exists.
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Question 1193867: Correction: An infinite geometric series has the first term u1 = a and u2 = (1/4a^2)-3a, where a > 0.
a. Find the values of a for which the sum to infinity of the series exists.
b. Find the value of a when s infinity = 76. Answer by math_tutor2020(3817) (Show Source):
u1 = first term = a
u2 = second term = (1/4)a^2 - 3a
r = common ratio
r = (u2)/(u1)
r = ((1/4)a^2 - 3a)/(a)
r = (a((1/4)a - 3))/(a)
r = (1/4)a - 3
where 'a' is nonzero, to avoid a division by zero error.
For the infinite geometric sum to converge to a finite value, we need -1 < r < 1 to be true.
The common ratio needs to be between -1 and 1, excluding both endpoints.
note1: I added 3 to all sides to undo the -3 in the middle
note2: I multiplied all sides by 4 to undo the 1/4
note3: The inequality signs do not flip for any of the steps
Answer: 8 < a < 16
The value 'a' needs to be between 8 and 16, excluding both endpoints, so that we converge to a finite sum.
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Part B
is the notation talking about the partial sum where n approaches infinity.
So informally we can say
I'll simply refer to it as S to make things a bit more simple.
This infinite sum S is only possible if 8 < a < 16 found back in part A earlier.
If that inequality holds up, then it leads to -1 < r < 1 and it allows us to use this infinite geometric sum formula
S = a/(1-r)
Let's plug in S = 76 and solve for 'a'
S = a/(1-r)
76 = a/(1-r)
76(1-r) = a
a = 76(1-r)
a = 76-76r
Now plug in r = (1/4)a - 3 and solve for 'a'
a = 76-76r
a = 76-76( (1/4)a-3 )
a = 76-19a + 228
a+19a = 76 + 228
20a = 304
a = 304/20
a = 76/5
a = 15.2
This value of 'a' satisfies the inequality 8 < a < 16, so this is a valid value to allow the infinite geometric series to converge to a finite sum.
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