Idon't know how to approach it, i've been looking at my notes and i can't grasp it.

  8

Step-by-step explanation:

You can skip directly to the formula for the sum of an infinite sequence with first term a₁ and common ratio r:

  S = a₁/(1-r)

Your values of the variables in this formula are a₁ = 6 and r = 2/8. Putting these into the formula gives ...

  S = 6/(1 -2/8) = 6/(6/8) = 8

The sum of the infinite geometric sequence is 8.

The above formula is the degenerate form of the formula for the sum of a finite sequence:

  S = a₁((rⁿ -1)/(r -1))

When the common ratio r has a magnitude less than 1, the term rⁿ tends to zero as n gets very large. When that term is zero, the sum of the infinite sequence is ...

  S = a₁(-1/(r-1)) = a₁/(1-r)

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step-by-step explanation:

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the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12