Question

Find f(5) for this sequence: (2 points)
f(1) = 2 and f(2) = 3, f(n) = f(1) + f(2) + f(n - 1), for n > 2.

f(5) = ______

Answer 1

\(\large\color{black}{ \displaystyle f(\color{blue}{3})=f(1)+f(2)+f(\color{blue}{3}-1) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{3})=f(1)+f(2)+f(2) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{3})=f(1)+2f(2) }\)

\(\large\color{black}{ \displaystyle f(\color{blue}{4})=f(1)+f(2)+f(\color{blue}{4}-1) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{4})=f(1)+f(2)+f(3) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{4})=f(1)+f(2)+f(1)+2f(2) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{4})=2f(1)+3f(2) }\)

\(\large\color{black}{ \displaystyle f(\color{blue}{5})=f(1)+f(2)+f(\color{blue}{5}-1) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{5})=f(1)+f(2)+f(4) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{5})=f(1)+f(2)+2f(1)+3f(2) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{5})=3f(1)+4f(2) }\)

Answer 2

the bottom line is:
\(\large\color{black}{ \displaystyle f(\color{blue}{3})=f(1)+2f(2) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{4})=2f(1)+3f(2) }\)
\(\large\color{black}{ \displaystyle f(\color{blue}{5})=3f(1)+4f(2) }\)

Answer 3

And this is a proof that d=f(1)+f(2)

Answer 4

so even if we not given f(1) and f(2) we can tell that d=f(1)+f(2),
and here it is even better, because we know f(1)=2 and f(2)=3
So
d=2+3=5

Answer 5

f(5)=3•2+4•3

Answer 6

thanks :)