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# infinite geometric series

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An infinite geometric series has a first term of 12 and a second term of 4.  A second infinite geometric series has the same first term of 12, a second term of 4 + n and a sum of three times that of the first series. Find the value of n.

Nov 22, 2020

#1
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1 - Sum up the first infinite series using the "Infinite Series formula"

2 - Whatever sum you get from (1) above, multiply it by 3 and that is the sum of the 2nd infinite series.

3 - Again, use the "infinite Sum Formula" to find n, bearing in mind that the "common ratio" of the 2nd series is:(4 + n) / 12. You should get n = 16 / 3

Nov 22, 2020
#2
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First series......common ratio between terms is  4/12 = 1/3

Sum of this series is  represented by :       12 / [ 1 -1/3 ] =  12/ (2/3)   = 18

The second series has a sum of  18 * 3  = 54

The common ratio is  [4 + n] /12

So

12 / [ 1 - (4 +n)/12 ]  =  54

12/54  =  1  - (4 + n)  /12

2/9 = 1  - (4 + n) /12

-7/9  = - (4 + n) /12

12 (7/9) = 4 + n

84/9 - 4   = n

48/9 = n  =  16/3  [just as the Guest found!!!  ]   Nov 22, 2020