Question

Find the arithmetic means in each sequence.

3, ?, ?, ?, ?, ? , 27

Short answer

The arithmetic means in the sequence are7, 11, 15, 19, 23.

Step-by-step solution

Step 1. Given Information.

Given there are five arithmetic means between 3 and 27.

Step 2. Calculation.

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

Since there are five arithmetic means between 3 and 27,

The first term is $a_1=3$

And the seventh term is $a_7=27$

Plugging the values in the formula:

$\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_7=a_1+(7-1)d\\ 27=3+6d\\ 6d=27-3\\ 6d=24\\ d=4\end{array}$

Hence the common difference is 4.

Since the first term in the arithmetic sequence is 3:

The next term in the arithmetic sequence is $3+4=7$

The next term in the arithmetic sequence is $7+4=11$

The next term in the arithmetic sequence is $11+4=15$

The next term in the arithmetic sequence is $15+4=19$

The next term in the arithmetic sequence is $19+4=23$

And the fourth term in the arithmetic sequence is 27.

Step 3. Conclusion.

Hence, the arithmetic means in the sequence are 7, 11, 15, 19, 23.