It is known that the sequence {an} is an arithmetic sequence, and the sequence AK1, ak2, ak3 ,akn,… It's just an equal ratio sequence, Where K1 = 1, K2 = 5, K3 = 17 1) Find kn; 2) Find the first n terms and TN of sequence {kn}

It is known that the sequence {an} is an arithmetic sequence, and the sequence AK1, ak2, ak3 ,akn,… It's just an equal ratio sequence, Where K1 = 1, K2 = 5, K3 = 17 1) Find kn; 2) Find the first n terms and TN of sequence {kn}

1) From (A5) ^ 2 = A1 (A17) to (a1 + 4D) ^ 2 = A1 (a1 + 16d), A1 = 2D is simplified
So A1 = 2D, A5 = 6D, A17 = 18D, the common ratio of the equal ratio sequence is 3
So AKN = 2D · 3 ^ (n-1) and AKN is the kn term of the arithmetic sequence, so
2D · 3 ^ (n-1) = 2D + (kn-1) d, then kn = 2 · 3 ^ (n-1) - 1
2)Tn=2·3^0-1+2·3^1-1+2·3^2-1+…… +2·3^(n-1)-1
=2·3^0+2·3^1+2·3^2+…… +2·3^(n-1)-n
3Tn=2·3^1+2·3^2+2·3^3+…… +2·3^(n-1)+2·3^n-3n
By subtracting the two formulas, 2tn = 2.3 ^ n-2n-2
So TN = 3 ^ n-n-1