Question

For these recurrence relations, solve for general equation using characteristics and particular. Use initial condition if given.

a. fn+1 = 1 Initial condition: fo = 2 b. fn+1 -fn-n=0 n-1 1+fi = fn+1 Initial conditions: fo = 1, f1 = 1, n > 1 i=0

Answer 1

q iching complimentary Sog (p). Given recurrence relation is, fint, -fn-n=0 that is final fuch and initial value ù not given, corresponding Homogeneous eon ist font-fn=o Characteristics equation is put put fn=qu Y ahtl_gn=o - anGa-D=0, but 970 fn solution is . fl)- Acion We try particular solution for as follows: PJ Bt Cn be particular solution Btc.(n+1) (e) fcp) en becomes [B+ c (nt)]-[B+ca] Btcnt. c - B-cnn can f (B) = B+ C² = Btn2 so General solution, fn=fice for (P) fn= AC + B + n² Let fr •CP) fat Then nti fnt is n-1 Given Recumence Relation is, it fi- frit fo=1, fi= 1 nolu fath fotfit--fn-, tl Replace n by ntl then we get fort tot fit finasti ti finte fo + fut fn ti foto f