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:
https://brainly.com/question/11559160
