Question

What’s the formula for the series -5,10,-20,40

Answer 1

Answer:

Stare at it long enough. You will notice that with each step you double the value, and change sign. It makes it a geometric sequence of ratio -2. At this point you have two ways of writing down:

Option 1: Ricorsively, the first term is -5, then the next term is -2 times the previous one:

$\left \{ {{a_0=-5} \atop {a_{n+1}=-2a_n}} \right.$

Option 2: In closed form, remembering that the n-th term in a geometric sequence is the starting term times the n-th power of the ratio:

$a_n=-5 (-2)^n$

Either work, depending on what you need to use it for.