Question

given an arithmetic sequence with a7 =19and a12 = 54, find a=15.

Answer 1

It's worth noting that elements in an arithmetic sequence just keep adding a certain number, over and over again, that number which you add is what's called the common difference of the sequence...

That said, the sequence is pretty much (but not entirely) defined by its 'common difference'

To get the common difference, given two elements \(\large a_m \) and \(\large a_n\), use this formula...
\[\Large d = \frac{a_n-a_m}{n-m}\]
Now, could you solve for d?
Use \[\large a_7 = 19\]and\[\large a_{12} =54\]

Answer 2

a7 = a0 + 7d
a12 = a0 + 12d
a12 - a7 = 12d - 7d = 5d
find d
then find a0
then
a15 = a0 + 15d