Question

Find the arithmetic means in each sequence.

55, ? , ? , ? , 115

Short answer

The arithmetic means in the sequence are70, 85, 100.

Step-by-step solution

Step 1. Given Information.

Given there are three arithmetic means between 55 and 115.

Step 2. Calculation.

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

Since there are three arithmetic means between 55 and 115,

The first term is $a_1=55$

And the fifth term is $a_5=115$

Plugging the values in the formula:

$\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_5=a_1+(5-1)d\\ 115=55+4d\\ 4d=115-55\\ 4d=60\\ d=15\end{array}$

Hence the common difference is 15.

Since the first term in the arithmetic sequence is 55:

The next term in the arithmetic sequence is $55+15=70$

The next term in the arithmetic sequence is $70+15=85$

The next term in the arithmetic sequence is $85+15=100$

And the fifth term in the arithmetic sequence is 115.

Step 3. Conclusion.

Hence, the arithmetic means in the sequence are 70, 85, 100.