Question

number 12 please!

In each of 11-16 suppose a sequence satisfies the given recurrence relation and initial conditions. Find an explicit formula for the sequence. 11. dx = 4dx-2, for each integer k 22 do = 1, d, = -1 12. ek = 9ek-2, for each integer k = 2 e0=0, es=2 13. r = 2rk-1 - k-2, for each integer k 2 2 ro = 1, rı = 4 14. Sk = - 45x-1 - 45k-2, for every integer k 2 2 so = 0, $ı = -1 15. tk = 6tk-1 - 9tk-2, for each integer k 2 2 to = 1, t1 = 3

Answer 1

-Anawea: For each integer k>r Griven e-ge The chanacteristic equation-for-the Sequence Tecurence elation is get We Dividing by t qt alebra! Basic 2-9-0 Ct+3) ct-)o [factoring

t3-0 )t-3-0 t3 01 t-3 Bthe distinctgoot theorem, the ualue of t obtained aboue for Some values of Safithe equalion - CCaD3, For all irdegent that e0 t is giuen n-o into euation, weget Salkatituting c. C39+D-0 badic algebma o[Ga-Ca C+D D-C t'S qfuern that e,= Sub stituting wegat n- ito equation O equating to,

3c+30= the ualues cf D obtained inequatior Sabstitat ing into equation get We 3C+3CC)=R -6C -Y's Sabatitading get The ualue ofC iwto equation (a), -3) D = and 0-3 .The explict fomuda for "he Sequence is en C-3) 3)-3 e +31-