Please help

1. Let the terms of the G.P. be a/r⁴,a/r³,a/r²,a/r,a,ar,ar²,ar³,ar⁴ These are the 9 terms in G.P. fifth term is a , a= 3√3 product of nine terms is a^9 = 1594323√3 There is no option in the answer
2. The n^th term is (3*n – 2)/5ⁿ

∑(3*n – 2)/5ⁿ  =   3*^inf∑_n=1(n/5ⁿ) -2*^inf∑_n=1(1/5ⁿ)

The sum of the second sigma is 5/4 (infinite G.P. 1/(1-1/5) ) let first sigma be denoted by S

S = ^inf∑_n=1(1/5ⁿ) + ^inf∑_n=1(n-1/5ⁿ) ( since n = (n-1) + 1 )

The first sigma in S is 5/4. Let the second sigma be S₁. Now S₁ = ^inf∑_n=1(n-1/5ⁿ) but notice that the first term of S₁ is 0 as n-1=0 if n=1 so S₁ can be written as ^inf∑_n=2(n-1/5ⁿ) now instead of n substitute t = n-1 in the equation of S₁. Now

              S₁ = ^inf∑_t=1(t/5^t+1)

Taking 1/5 as common

               S₁ =1/5( ^inf∑_t=1(t/5^t)) = S/5 (notice that the expression in brackets is same as the original S)

so S = 5/4 + S/5  implies 4S/5 = 5/4 implies S = 25/16

substituting in the first equation we get

sum = 3*25/16 – 2*5/4 = 35/16

In the above answer  for the second question given by me the answer was correct but the method with which I solved the sum is wrong. My first mistake was that the term t_n is (3*n-2)/5^n-1. Now to sum it…
∑(3*n – 2)/5^n-1 =15*^inf∑_n=1(n/5ⁿ) -10*^inf∑_n=1(1/5ⁿ)

My second mistake was to say that the sum of second sigma is 5/4. It is not 5/4 but it is 1/4 .

S₀ = 1/5(1/(1 – 1/5)) = ¼ (I forgot that the initial term is 1/5 and not 1)

let first sigma be denoted by S

S= ^inf∑_n=1(1/5ⁿ)+ ^inf∑_n=1(n-1/5ⁿ)(since n = (n-1)+1)

The first sigma is S is ¼ Let the second sigma be S₁. Now S₁ = ^inf∑_n=1(n-1/5ⁿ) but notice that the first term of S₁ is 0 as n-1=0 if n=1 so S₁ can be written as ^inf∑_n=2(n-1/5ⁿ) now instead of n substitute t = n-1 in the equation of S₁. Now

              S₁ = ^inf∑_t=1(t/5^t+1)

Taking 1/5 as common S₁ =1/5( ^inf∑_t=1(t/5^t)) = S/5 (notice that the expression in brackets is same as the original S)

So S = 1/4 + S/5 so 4S/5 = 1/4 implies S = 5/16

Now the sum is 15*5/16 – 10/4 = 75/16 – 40/16 = 35/16

I was very very lucky that I got the right answer.