Find explicit formulas for each of the following sequences: . (a) an=4+3an-1, do 6 (b) an 2an-1 +3an-2, ao = 4, a₁ = 9 =

What is an nth partial sum?

Need help with this problem. It states:   Determine the number of n-digit quaternary (0, 1, 2, 3) sequences (in closed form) where there is never a 3 anywhere to the right of a 0.   My current solution: I split it up into two parts, one for if there is a 0 in the sequence and one without. I've gathered that there are 3^(n-1) possibilities if the first digit is a 0 and you add it with the remaining which is 3^n? But something seems missing. Please help!!!   Also how should I set up my characteristic polynomial? (S(k))? I have S(k) = S(k-1) + k*3S(k-1) and trying to solve for the coefficients. But it doesn't seem to be quite right. Please help and thank you!

A sequence {a^n} of real numbers is defined recursively by a_1=2 for n ≥ 2, a_n=(2+1∙a_1^2+2∙a_2^2+⋯+(n-1) a_(n-1)^2)/n Determine a_(2,) a_3, and a_4. Show all work. Clearly, a_n is a rational number for each n ϵ N. Based on the information in a), however, what conjecture does this suggest?