find-the-sequence-u-n-wich-verify-u-n-2-u-n-1-1-2-n-

Question Number 26400 by abdo imad last updated on 25/Dec/17

f i n d t h e s e q u e n c e (u_{n}) w i c h v e r i f y u_{n} - 2 u_{n - 1} + 1 = 2^{n}

Answered by prakash jain last updated on 25/Dec/17

x - 2 = 0

\Rightarrow x = 2

h o m o g e n o u s s o l u t i o n = c 2^{n}

p a r t i c u l a r s o l u t i o n

u_{n} - 2 u_{n - 1} = 2^{n} - 1

Here we will solve for 2 parts

part I

u_{n} - 2 u_{n - 1} = - 1 \Rightarrow u_{n} (part 1) = 1

part II

u_{n} - 2 u_{n - 1} = 2^{n}

u_{n} = a n 2^{n}

a n 2^{n} - a (n - 1) 2^{n} = 2^{n}

\Rightarrow a = 1

p a r t i c u l a r s o l u t i o n

u_{n} = n 2^{n} + 1

u_{n} = c 2^{n} + n 2^{n} + 1