Question

Find the arithmetic means in each sequence.

10, ? , ? , -8

Short answer

The arithmetic means in the sequence are4, -2.

Step-by-step solution

Step 1. Given Information.

Given there are two arithmetic means between 10 and -8.

Step 2. Calculation.

The n^th term of an arithmetic sequence is given by $a_n=a_1+(n-1)d$.

Since there are two arithmetic means between 10 and -8,

The first term is $a_1=10a$

And the fourth term is $a_4=-8$

Plugging the values in the formula:

$\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_4=a_1+(4-1)d\\ -8=10+3d\\ 3d=-8-10\\ 3d=-18\\ d=-6\end{array}$

Hence the common difference is -6.

Since the first term in the arithmetic sequence is 10:

The next term in the arithmetic sequence is $10-6=4$

The next term in the arithmetic sequence is $4-6=-2$

And the fourth term in the arithmetic sequence is -8.

Step 3. Conclusion.

Hence, the arithmetic means in the sequence are 4, -2.