1. We have two geometric sequences of positive real numbers: $$6,a,b\text{ and }\frac{1}{b},a,54$$Solve for $a$.
2. 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 four times that of the first series. Find the value of $n.$
2)
4/12 = 1/3 Common ratio of the first GS
Sum = F / [1 - R], where F= First term, R = Common ratio
Sum =12/ [1 - 1/3]
Sum =12 / (2/3)
Sum = 12 x 3/2
Sum = 36/2
Sum = 18 - Sum of the first GS
18 x 4 = 72 - Sum of the second GS
72 = 12 / [1 - R] divide both sides by 12
6 = 1 / [1 - R] cross multiply
6 [1 - R] = 1 divide both sides by 6
[1 - R] = 1/6 subtract 1 from both sides
- R = 1/6 - 1
- R = - 5/6 Multiply both sides by -1
R = 5/6 - Common ratio of the second GS
12 x 5/6 = 60 / 6 =10 - this is the 2nd term of the second GS
10 - 4 = 6 value of n of the 2nd term of the second GS.
1) Sorry, Can't read the LaTex of your first question.
