Determine the characteristic equation for an 2an-1 what

Question

Question: 1. Determine the characteristic equation for a_n = 2a_n-1. What are its roots? Using this information and the initial condition a₁ = 42, determine a closed form formula.

2. For each of the following recurrences, decide whether one could use (i) kth differences, (ii) the characteristic equation, or (iii) neither in order to find a closed form for the recurrence.

(a) a_n = 2a_{n - 1} + 2

(b) a_n = a_{n - 1} + 2a_{n - 2} + 3a_{n - 3 +} 4a_{n - 4}.

(c) a_n = 3a_{n - 3} + 3n

(d) a_n = a_{n - 1} - 2n².