If f(n)=2f(n-1)+3n and f(1) =-2 what do the next five terms equal.

Answer 1

Hi, Logan --

In general, f(), read "f of", is the name of the function that tells us how to find each new value based on the previous value.

But f(1) = –2, read "f of one equals negative two", tells us that the starting value (value #1) is negative two.

To find f(2), "f of two", that is, value #2, first plug 2 in for n in the formula. Remember that 2f(n – 1) means 2·f(n – 1) and 3n means 3·n.

f(n) = 2·f(n – 1) + 3·n

f(2) = 2·f(2 – 1) + 3·2

= 2·f(1) + 6

Now use what we already know, namely f(1) = –2. So:

f(2) = 2·(–2) +6

= –4 + 6

= 2

So the second value is 2.

Continue the same way. To find the third value, let n = 3.

f(n) = 2·f(n – 1) + 3·n

f(3) = 2·f(3 – 1) + 3·3

= 2·f(2) + 9

= 2·2 + 9, since f(2) = 2

= 4 + 9

= 13

So the third value is 13.

Here's what we get if we keep going:

n f(n) = 2·f(n – 1) + 3·n

---------------------------------

1 –2

2 2·f(1) + 3·2 = 2·(–2) + 6 = –4 + 6 = 2

3 2·f(2) + 3·3 = 2·2 + 9 = 4 + 9 = 13

4 2·f(3) + 3·4 = 2·13 + 12 = 26 + 12 = 38

5 2·f(4) + 3·5 = 2·38 + 15 = 76 + 15 = 91

6 2·f(5) + 3·6 = 2·91 + 18 = 182 + 18 = 200

or

n f(n)

-----------

1 -2

2 2

3 13

4 38

5 91

6 200

The next five values after –2 are 2, 13, 38, 91, and 200.

Answer 2

Remember that f(n-1) is the term right before f(n). We need to know the previous term to be able to evaluate the equation. Since we don't know the term before f(1), let's look at the term after it: f(2). This is possible because we know the term before it, it's f(1) and it equals -2!

So let's try f(2):

f(2) = 2f(2-1) + 3(2)

= 2f(1) + 3(2)

You can do the rest!

Answer 3

When you put x=1 , f(1) = 2f(0) + 3 = -2 => f(0) = -2.5

f(2) = 2f(1) + 6 = +2

f(3) = 2f(2) + 9 = 13

f(4) = 2f(3) + 12 = 38

f(5) = 2f(4) + 15 = 91

f(6) = 2f(5) + 18 = 200