How do you find the arithmetic means of the sequence -8, __, __, __, __, 7?

Question

Serenity Jones · Accepted Answer

For problems like these, we have to ask ourselves "How many times do I have to add the common difference, #d#, to the first term to get the last term?".

To answer, put a "+d" between -8 and the first space, between the first space and the second space, continuing until you add a d between the last space and the #7#.

In the end, you should have #5# d's.

We can set up the following equation:

#-8 + 5d = 7#

#5d = 15#

#d = 3#

I'm assuming that by arithmetic means you mean average, or mean, of the whole sequence.

#m = (-8 + -5 + -2 + 1 + 4 + 7)/6#

#m = -3/6#

#m = -1/2#

Hence, the mean of the numbers in the sequence is #-1/2#.

Hopefully this helps!

Serenity Johnson · Answer

undefined To find the arithmetic mean of a sequence, you sum all the numbers in the sequence and then divide by the total count of numbers. In this case, we have -8 and 7, and four missing numbers in between. Since the sequence is evenly spaced, we can calculate the missing numbers by finding the common difference (d) and using it to fill in the sequence.

First, find the common difference (d) using the formula:
$$ d = \frac{{\text{{Last term}} - \text{{First term}}}}{{\text{{Number of terms}} - 1}} $$

$$ d = \frac{{7 - (-8)}}{{6}} = \frac{{15}}{{6}} = \frac{5}{2} $$

Now, fill in the missing numbers using the common difference:
$$ -8, -8 + \frac{5}{2}, -8 + 2\left(\frac{5}{2}\right), -8 + 3\left(\frac{5}{2}\right), -8 + 4\left(\frac{5}{2}\right), 7 $$

$$ -8, -8 + \frac{5}{2}, -8 + 2\left(\frac{5}{2}\right), -8 + 3\left(\frac{5}{2}\right), -8 + 4\left(\frac{5}{2}\right), 7 $$

$$ -8, -8 + \frac{5}{2}, -8 + 5, -8 + \frac{15}{2}, -8 + 10, 7 $$

$$ -8, -8 + \frac{5}{2}, -3, -8 + \frac{15}{2}, 2, 7 $$

Now, sum all the numbers and divide by the total count of numbers:
$$ \frac{{-8 + (-3) + (-8 + \frac{15}{2}) + 2 + 7}}{6} $$

$$ \frac{{-8 - 3 - 8 + \frac{15}{2} + 2 + 7}}{6} $$

$$ \frac{{-12 + \frac{15}{2} + 6}}{6} $$

$$ \frac{{-12 + 7.5 + 6}}{6} $$

$$ \frac{{1.5}}{6} $$

$$ \frac{{3}}{12} $$

$$ \frac{1}{4} $$

So, the arithmetic mean of the sequence is $ \frac{1}{4} $.