There are ‘n’ arithmetic means between 33 and – 3. If second last mean: second mean = 1: 5, find the value of n.

We know that:

There are n arithmetic means between the numbers 33 and -3

second last mean: second mean = 1:5

Using this, we can find that the value of n is 9.

Now let's see how we need to use the information.

"There are n arithmetic means between the numbers 33 and -3"

This means that we will have a sequence like:

33, a₁, a₂, ..., aₙ, -3

And by the given ratio, we know that:

aₙ₋₂:a₂  = 1:5

Also, because this is an arithmetic sequence we have:

a₁ = 33 + d

a₂ = 33 + d + d

aₙ ₋ ₂ = 33 + (n - 2)*d = -3 - 2*d

Because of the ratio, we will have that:

aₙ₋₂/a₂ = 1/5

We can replace what the left side by:

(-3 - 2*d)/(33 + 2*d) = 1/5

now we can solve this for d.

-3 - 2*d = (33 + 2*d)*(1/5)

5*(-3 - 2*d ) = (33 + 2*d)

-15 - 10d = 33 + 2d

-15 - 33 = 2d + 10d

-48 = 12d

-48/12 = -4 = d

So now we know that the value of d is -4.

Now we can use the equation

aₙ ₋ ₂ = 33 + (n - 2)*d = -3 - 2*d

to find the value of n:

33 + (n - 2)*d = -3 - 2*d

33 + (n - 2)*-4 = -3 - 2*-4

33 - 4n + 8 =  -3 + 8

41 - 4n = 5

-4n = 5 - 41 = -36

n = 36/4 = 9

The value of n is 9.

If you want to learn more, you can read:

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