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What is the smallest integer that can possibly be the sum of an infinite geometric series whose first term is 13?

 Mar 26, 2023
 #1
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The sum of an infinite geometric series is given by the formula:

S = a/(1-r)

where 'a' is the first term of the series and 'r' is the common ratio.

For the given series, the first term is 13. To find the smallest possible sum, we need to find the smallest possible value of 'r' such that the series converges.

For a series to converge, the absolute value of the common ratio must be less than 1. That is:

|r| < 1

Let's consider two cases:

Case 1: r is positive In this case, the smallest possible value of 'r' that satisfies the condition |r|<1 is r=1/2. Therefore, the sum of the infinite geometric series is:

S = a/(1-r) = 13/(1-1/2) = 26

Case 2: r is negative In this case, the smallest possible value of 'r' that satisfies the condition |r|<1 is r=-1/2. Therefore, the sum of the infinite geometric series is:

S = a/(1-r) = 13/(1+1/2) = 8.6666...

So the smallest possible integer that can be the sum of an infinite geometric series with a first term of 13 is 26.

 Mar 26, 2023
 #2
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thanks, but its wrong. 

Guest Mar 26, 2023
 #3
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Thanks guest answerer,

a=13

You are certainly right that |r|<1

 

\(S_{\infty}=\frac{a}{1-r}\\~\\ S_{\infty}=\frac{13}{1-r}\\~\\\)

Now for this question  \(S_{\infty}\)    must be an integer so I will let it be N

r must be between -1 and 1   which means that  1-r  is between 0 and 2

so N is between (not including)  13/2   and  infinity

So the smallest is infinitesimally bigger than 6.5   

Which means ths smallest integer value is 7

 

\(S_{\infty}=\frac{13}{1-r}\\~\\ N=\frac{13}{1-r}\\~\\ 7=\frac{13}{1-r}\\~\\ 7(1-r)=13\\~\\ -7r=6\\~\\ r=\frac{-6}{7} \)

 Mar 27, 2023
 #4
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Thank you!

Guest Mar 29, 2023
 #5
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ssume that we have two variables a and b, and that: a = b Multiply both sides by a to get: a2 = ab Subtract b2 from both sides to get: a2 – b2 = ab – b2 This is the tricky part: Factor the left side (using FOIL from algebra) to get (a + b)(a – b) and factor out b from the right side to get b(a – b). If you’re not sure how FOIL or factoring works, don’t worry—you can check that this all works by multiplying everything out to see that it matches. The end result is that our equation has become: (a + b)(a – b) = b(a – b) Since (a – b) appears on both sides, we can cancel it to get: a + b = b Since a = b (that’s the assumption we started with), we can substitute b in for a to get: b + b = b Combining the two terms on the left gives us: 2b = b Since b appears on both sides, we can divide through by b to get: 2 = 1
 Apr 4, 2023

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