Thread: Recurrence Relation

Solve the given recurrence relation with for the initial conditions given.
a_n = 7a_(n-1) - 10a_(n-2)
a_0 = 5
a_1 = 16

I am missing something and cannot come up with the answer here.

Pwr

Hello, pwr_hngry!

Solve the given recurrence relation with for the initial conditions given.
an .= .7an-1 - 10an-2, .a0 = 5, .a1 = 16

The characteristic equation is: .r² - 7r + 10 .= .0

. . which factors: .(r - 2)(r - 5) .= .0

. . and has roots: .r .= .2, 5

So we have: .an .= .b·2^n + c·5^n


From a0 = 5, we have: .b·2^0 + c·5^0 .= .5 . . → . . b + c .= .5

From a1 = 16, we have: .b·2^1 + c·5^1 .= .16 . . → . . 2b + 5c .= .16

Solve the system of equations and get: .b = 3, .c = 2


Therefore: .an .= .3·2^n + 2·5^n