An animal use 50% of its fat every night, but is able to add $4$ grams to its fat reserve every day. This same animal begin with 3 grams of fat on the first night.
Derive a model for its fat reserve at the beginning of $(t-1)$ night.
I tried to write down first couple nights and find a relation.
Night 1: $3$
Night 2: $3*0.5+4$
Night 3: $(3*0.5+4)*0.5 + 4$
Night 4: $((3*0.5+4)*0.5 + 4)*0.5 + 4$
I feel like I am onto something, but I can't quite get it. Maybe it's my lack of understanding of recursive relationship. I need some help on this.
Since you're trying to generalize a pattern, it'll help to have notation for the items in the sequence. For every natural number $t$, let's define $x_t$ to be the fat amount on the beginning of the $t$th night. You've calculated $x_1,x_2,x_3,x_4$. Could you write a formula for $x_{100}$ in terms of $x_{99}$? A formula for $x_{t+1}$ in terms of $x_t$? Then you're done. (It looks like you're not required to solve the recurrence relation, just write it down.)