Solve these recurrence relations together with the initial conditions given an+2=−4an+1+5an for n≤0, a0=2, a1=8

Question

Solve these recurrence relations together with the initial conditions given  an+2=−4an+1+5an for n≤0, a0=2, a1=8

Fearen · Accepted Answer

The given reccuranse relations is   an+2=−4an+1+5an for n≤0, a0=2, a1=8  therefore the characteristic equation for the reccurence solution is  r2+4r−5=0  ⇒r2+(5−1)r−5=0  ⇒r2+5r−r−5=0  ⇒r(r+5)−1(r+5)=0  ⇒(r+5)(r−1)=0  ⇒r=−5,1  therefore, characteristic roots are 1, -5.  so the solution of the given recurrence relation toill be  an=A(−5)n+B(1)n  ⇒an=A(−5)n+B  Now a0=2  so from equation  a0=A(−5)0+B  ⇒a0=A+B  ⇒A+B=2  as a1=8, therefore from  a1=A(−5)1+B  ⇒8=−5A+B  ⇒5A+B=8  Sunstracting equation we get  A+B+5A−B=2−8  ⇒6A=−6  ⇒A=−66=−1  ⇒A=1  Therefore,   −1+b=2⇒B=2+1=3  ⇒B=3  Putting these values into equation the solution of given recurrence relation is  an=−(−5)n+3

Solve these recurrence relations together with the initial conditions given a_{n+2}=-4a_{n+1}+5a_n

Answered question

Answer & Explanation

New Questions in Algebra I