Find the first arithmetic mean between 100 and 135 given the following sequence:

..., 100, ___, ___, ___, ___, 135, ...

so we are looking at 6 terms

sum(6) = 3(first + last) = 705
mean of those 6 numbers =705/6 = 117.5

128

To find the first arithmetic mean between 100 and 135 in the given sequence, we need to determine the missing terms.

The arithmetic mean between two numbers is the sum of the numbers divided by the number of terms. In this case, we have two known terms (100 and 135) and four unknown terms.

Let's label the missing terms as A, B, C, and D.

The formula for the arithmetic mean is:

mean = (sum of terms) / (number of terms)

In this case, the number of terms is 6 since we have 6 terms in the sequence (including the missing terms). So, we can write the equation:

mean = (100 + A + B + C + D + 135) / 6

To find the first arithmetic mean, we need to make sure that all four missing terms are equal. Let's assume all four are equal to a variable, which we will call X.

Now, the equation becomes:

mean = (100 + X + X + X + X + 135) / 6

Simplifying:

mean = (100 + 4X + 135) / 6

mean = (235 + 4X) / 6

To find the value of X, we need to equate the mean with the desired value.

Let's assume the first arithmetic mean is M. So, our equation becomes:

M = (235 + 4X) / 6

To solve for X, we can rearrange the equation:

235 + 4X = 6M

4X = 6M - 235

X = (6M - 235) / 4

Now, we substitute this value of X back into the equation to find the first arithmetic mean.

mean = (100 + X + X + X + X + 135) / 6

mean = (100 + X + X + X + X + 135) / 6

mean = (100 + (6M - 235) / 4 + (6M - 235) / 4 + (6M - 235) / 4 + (6M - 235) / 4 + 135) / 6

Simplifying this expression will give us the value of the first arithmetic mean between 100 and 135 in the given sequence.