Question

Sort the arithmetic means (AM) of the given APs in ascending order if each AP consist of 11 terms.

• A. 1, 3, 5,...
• B. -10, -5, 0,...
• C. -2, 0, 2,...
• D. 12, 12, 12,...

Answer 1

The arithmetic mean of an AP is given by
$⟹AM=\frac{\text{Sum of all terms}}{\text{Number of terms in the AP}}.$

If $a$ and $d$ are the first term and the common difference of the AP respectively, then we have
$AM=\frac{S_n}{2}=\frac{\frac{n}{2}\times (2a+(n-1\left)d\right)}{n}=\frac{2a+(n-1)d}{2},$
where $S_n$ denotes the sum to $n$ terms of the AP.
Given that $n=11$ for all the APs.
In the AP 1, 3, 5,...
$a=1$ and $d=a_2-a_1=3-1=2.$
$\therefore AM=\frac{2a+(n-1)d}{2}$
$=\frac{2\left(1\right)+(11-1)2}{2}$
$=11$

In the AP -10, -5, 0,...we have $a=-10$ and $d=5.$

$\therefore AM=\frac{2a+(n-1)d}{2}$
$=\frac{2(-10)+(11-1)5}{2}$
$=15$

Using the exact same procedure, we get AM = 8 for the AP -2, 0, 2,... and AM=12 for the AP 12, 12, 12,...

Hence, AM of -2, 0, 2,... < AM of 1, 3, 5,... < AM of 12, 12, 12,... < AM of -10, -5, 0,...