If f(1)=3 and f(n)=f(n−1)+3 then find the value of f(5).

Answer 1

f(n) = f(n-1) +3

f(1) = f(1-1)+3 = f(0)+3= 3

f(0) = 0

f(1)=0+3=3

f(2)=3+3=6

f(3)=6+3=9

f(4)=9+3=12

f(5) =12+3 =15

f(5)=15

f(n)=3n

f(5)=3(5) = 15

f(4)=3(4) = 12

f(3) =3(3) =9

f(2)=3(2) = 6

f(1) =3(1) = 3

f(0) =3(0) = 0

Answer 2

f(n) = f(n - 1) + 3 defines an arithmetic sequence with d = 3

f(n) = f(1) + (n - 1)(3), this is the explicit definition

f(5) = 3 + 4(3)

f(5) = 3 + 12

f(5) = 15

IMHO using a_n = a_n-1 + 3 is a less confusing notation. f(x) should be reserved for functions (yes, sequences are functions, I know).

Answer 3

This is the recursive definition of a sequence:

f(1)=3

f(n)=f(n−1)+3 ... for n>1

The problem asks for the value of f(5). Note that 5 is a small value, so the value of f(5) may be found by finding f(2) thru f(5). However, recursion usually means starting with f(5) [a recursive case] and elaborating until f(1) is used [the base case]. Let's do that (note: this is one of the main uses of a STACK data structure).

With recursion, it is important to use the recursive case and the base case !!

f(5) = f(4) + 3

f(5) = ( f(3) + 3 ) + 3

f(5) = ( ( f(2) + 3 ) + 3 ) + 3

f(5) = ( ( ( f(1) + 3 ) + 3 ) + 3 ) + 3

f(5) = ( ( ( 3 + 3 ) + 3 ) + 3 ) + 3

f(5) = ( ( 6 + 3 ) + 3 ) + 3

f(5) = ( 9 + 3 ) + 3

f(5) = 12 + 3

f(5) = 15