# How do you find the arithmetic means of the sequence 55, __, __, __, 115?

Jan 6, 2017

Arithmetic Mean = $\textcolor{g r e e n}{85}$

#### Explanation:

Suppose the common difference is $d$
Then the first 5 terms are
[1] $a = 55$
[2] $a + d = 55 + d$
[3] a+2d=55+2dcolor(white)("XX")larr "middle term"
[4] $a + 3 d = 55 + 3 d$
[5] $a + 4 d = 55 + 4 d = 115$

Since $55 + 4 d = 115$
$\rightarrow 4 d = 60$

$\rightarrow d = 15$

The arithmetic means (average) of an arithmetic sequence with an odd number of terms is the value of the middle term.
In this case,
$\text{Average} = a + 2 d = 55 + 2 \cdot 15 = 55 + 30 = 85$