Question

Find a recursive formula for the sequence: 1, -1, -7, -25

Answer 1

Answer:

$a_n=3a_{n-1}-4$

Step-by-step explanation:

Try the answers

You can try the answers to see what works. You can expect all of the choices to match the first two terms, so try some farther down. Let's see if we can get -25 from -7.

a) 3*(-7) -4 = -21 -4 = -25 . . . . this one works

b) -7 -2 = -9 . . . . ≠ -25

c) -3(-7) +2 = 21 +2 = 23 . . . . ≠ -25

d) -2(-7) +1 = 14 +1 = 15 . . . . ≠ -25

The formula that works is the first one.

_____

Derive it

All these formulas depend on the previous term only, so we can write equations that show the required relationships. Let the unknown coefficients in our recursion formula be p and q, as in ...

$a_n=p\cdot a_{n-1}+q$

Then, to get the second term from the first, we have

... 1·p +q = -1

And to get the third term from the second, we have

... -1·p +q = -7

Subtracting the second equation from the first gives ...

... 2p = 6

... p = 3 . . . . . . . this is sufficient to identify the first answer as correct

We can find q from the first equation.

... q = -1 -p = -1 -3 = -4

So, our recursion relation is ...

$a_n=3a_{n-1}-4$