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find b if b/7 + b^2/7^2 + b^3/7^3...=7.

 May 26, 2023
 #1
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The given equation is a geometric series with first term b and common ratio 71​. The sum of a geometric series is given by the formula S=a​(1−rn)/(1-r)​, where a1​ is the first term, r is the common ratio, and n is the number of terms. In this case, we have S=7, a_1​=b, and r=1/7​.

Solving for n, we get n=−log(1−r)/log(r)​=log(1/7)/log(7)​=−log(7)/log(7)​=−1. Substituting this value of n into the formula for the sum of a geometric series, we get S=a(1-r^n)/(1-r)​=b(1-(1/7)^(-1))/(1-1/7)​=b(1−7)/(-6)​=7. Solving for b, we get b=49/6​​.

 May 26, 2023
 #5
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Thanks, that's right!

Guest May 28, 2023
 #2
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sumfor(b, 1, 1000, (6.125^b / (7^b))==7

 

b ==6 + 1/8 ==6.125

 

These are the first 10 terms of GS:

 

(7 / 8, 49 / 64, 343 / 512, 2401 / 4096, 16807 / 32768, 117649 / 262144, 823543 / 2097152, 5764801 / 16777216, 40353607 / 134217728, 282475249 / 1073741824)

 May 26, 2023
edited by Guest  May 26, 2023
 #3
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The given equation is a geometric series with first term b and common ratio 1/7​. The sum of an infinite geometric series is equal to a/(1-r)​, where a is the first term and r is the common ratio. In this case, the sum is equal to 7, so we have:

b/7 + b^2/7^2 + b^3/7^3 + ... = 7

\frac{b}{1-\frac{1}{7}} = 7

\frac{b}{6} = 7

b = \(\boxed{42}\)

 May 26, 2023
 #4
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Sum=F / [1 - r]

Sum=42 / [1 - 1/7]

Sum=42 / (6/7) =42 x 7/6=49 - your solution.

 

Sum =(7/8) / [1 - 7/8]

Sum =(7/8) / [1/8]==7/8 x 8/1==56 /8 ==7 - solution above yours.

Guest May 26, 2023

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