Question

What is the eighth term given the sequence: +896, -448, +224,-112,....... ?

Answer 1

the eighth term is `" "-7`

Explanation:

Let any position count be `i`
Let any term be `a_i`

So
`a_1->" first term"`
`a_2->" second term"`
`a_i->" "ith" term"`

Two things I notice:
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Point 1:")->`Alternates between positive and negative starting at positive.

So this can be achieved by

`a_i->a_1 xx(-1)^2`

`a_i->a_2xx(-1)^3`

`a_i->a_3xx(-1)^4`

`=>a_i->a_ixx(-1)^(i+1)`

,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Point 2:")->`Each term is half the previous

So we have

`(-1)^2xxa_1`
`a_2=(-1)^3xx1/2xxa_1`
`a_3=(-1)^4xx1/2xx1/2xxa_1`

This implies `a_i=(-1)^(i+1)xx(1/2)^(i-1)xxa_1`

Where `a_1=+896`
,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)(a_i=896(-1)^(i+1)(1/2)^(i-1)`

Thus

`" "color(purple)(bar(ul(|color(white)(2/2)a_8=896(-1)^(8+1)(1/2)^(8-1) = -7color(white)(2/2)|)))`