If f(1)=6 and f(n)=-4(n-1) then find the value of f(6)

Answer 1

If you wrote only f(n)=-4(n-1),

the answer is simple: f(6) = -4(6-1) = -4(5) = -20

However, the problem likely is: f(1)=6 and f(n) = -4 * f(n-1)

This is a recursive definition of a linear function f:

f(1) = 6 that is, f₁ = 6

f(n) = -4 * f(n-1)), that is f_n = -4 * f_n-1

So, what is f(6) ?

f(6) = -4 * ( f(5) )

= -4 * ( -4 * ( f(4) )

= -4 * ( -4 * ( -4 * f(3) ) )

= -4 * ( -4 * ( -4 * ( -4 * f(2) ) ) )

= -4 * ( -4 * ( -4 * ( -4 * ( -4 * f(1) ) ) ) ) <****

= -4 * ( -4 * ( -4 * ( -4 * ( -4 * 6 ) ) ) )

= -4 * ( -4 * ( -4 * ( -4 * ( -24) ) ) )

= -4 * ( -4 * ( -4 * ( +96 ) ) )

= -4 * ( -4 * ( -384 ) )

= -4 * ( +1436 )

= -6144

To translate the recursive function definition to an explicit function definition,

f(n) = (-4)^n-1 * 6 [you may check line <*** above]

So, to quickly check your answer:

f(6) = (-4)⁵* 6

f(6) = (-1024)*6

f(6 = - 6144