Arithmetic means

If there are $7$ arithmetic means between $3$ and $35$ , the arithmetic progression has $9$ terms with $a_1=3$ and $a_9=35$ . In finding the arithmetic means between two terms, we can use the $n^{th}$ term formula.

$a_n=a_1+(n-1)d$

$35=3+(9-1)d$

$35=3+8d$

$32=8d$

$4=d$

It follows that $a_2=3+4=7;a_3=7+4=11;a_4=11+4=15;a_5=15+4=19;a_6=19+4=23;a_7=23+4=27;a_8=27+4=31$

Adding the digits of all the terms, we get a sum of $54$ .