Question(1):A G.P has a common ratio of 2 find the value of 'n' for which the sum 2n terms is 33 times the sum of n
S_n = a(r^n - 1)/(r-1)
so, that means you have
S_2n = a(2^(2n)-1)
S_n = a(2^n-1)
so,
a(2^(2n)-1) = 33*a(2^n-1)
2^(2n)-1 = 33*2^n - 33
2^(2n) - 33*2^n + 32 = 0
(2^n-1)(2^n-32) = 0
2^n = 1 or 32
n = 0 or 5
check:
GP: 1,2,4,8,16,32,64,128,256,512,...
S5 = 31
S10 = 1023
S10 = 33*S5