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Question 1190304: Please help me solve this math problem:If the sum of 1+2x+4x²+8x³+... 1¼. Find x
Found 3 solutions by ikleyn, math_tutor2020, greenestamps:
Answer by ikleyn(52788)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
first term = 1
second term = 2x = 2x*(First term)
third term = 4x^2 = 2x*2x = 2x*(second term)
fourth term = 8x^3 = 2x*4x^2 = 2x*(third term)
nth term = 2x*(term just before nth term)
As you can see, we have a geometric sequence with:
a = 1 = first term
r = 2x = common ratio
We start at 1, and each time we need a new term, we multiply by 2x.
If we want the infinite geometric series of 1+2x+4x^2+8x^3+... to converge to some finite value, then we need -1 < r < 1 to be true.
r must be between -1 and 1, excluding both endpoints.
Why? because we need to add smaller and smaller pieces in order to slowly approach the finite sum.
Otherwise, the sum will diverge and blow up to plus or minus infinity.
Or it may bounce around never settling on any value at all.
-1 < r < 1
-1 < 2x < 1
-1/2 < x < 1/2
-0.5 < x < 0.5
This establishes the boundaries of what x can be.
If -1 < r < 1 is true, then we can use this summation formula
S = a/(1-r)
where S in this case is the result of adding up the infinitely many terms of the geometric series.
Plug in the first term a = 1, the common ratio r = 2x, and the desired sum S = 1 & 1/4 = 1 + 1/4 = 1.25 and solve for x.
S = a/(1-r)
1.25 = 1/(1-2x)
1.25(1-2x) = 1
1.25-2.5x = 1
1.25-1 = 2.5x
2.5x = 0.25
x = 0.25/2.5
x = 25/250
x = (25*1)/(25*10)
x = 1/10
x = 0.1
Now let's go back to -0.5 < x < 0.5
Notice that x = 0.1 is in this interval, so we have satisfied the criteria needed for an infinite geometric sum.
Answer: x = 1/10 or x = 0.1
Answer by greenestamps(13200)  (Show Source):
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