In a certain sequence of numbers, a1, a2, a3, ..., an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

We have A.P. ($a_1$ , $a_2$ , $a_3$ ........ $a_n$),  
           
    Average of first m term = $\frac{Sum of first m terms}{m}$

$\frac{{\displaystyle \frac{m}{2}}\left[2a_1 + \left(m - 1\right)d\right]}{m}$ = m

$\Rightarrow$[2a​${}_1$ + (m - 1) (a${}_2$ - ​a${}_1$)] = 2m

here a${}_1$ = 1 we get
$\Rightarrow$[2 + (m - 1) (a${}_2$ - 1) ] = 2m
$\Rightarrow$[ 2 + ​a${}_2$ m - ​a${}_2$  - m + 1] = 2m

$\Rightarrow$m = $\frac{3 - a_2}{3 -a_2 }$
for finding value of a${}_{10}$ , put m = 10 and get 

$\Rightarrow$ 10 = $\frac{3 - a_2}{3 -a_2 }$
$\Rightarrow$ ​a${}_2$  = $\frac{27}{9}$ = 3
so the  common difference d = 3 - 1 =  2
For finding a${}_{10}$ we use formula of nth term of A.P. 
a_n = a + (n - 1) d

​$\Rightarrow$a${}_{10}$ = 1 + ( 10 - 1)2
$\Rightarrow$a${}_{10}$ = 19