Please with arithmetic equations?

Plug numbers into formulas you should know.

"her salary increased every year, following an arithmetic progression,"
an = a1 + (n-1)d. That describes any arithmetic progression.

"and that it took 13 years for her salary to double that in her first year."
a13 = a1 + (13 - 1)d = 2a1
a1 + 12d = 2a1
-a1 + 12d = 0

"Given that the graduate's total earnings for her first 10 years with the firm was $1,650,000"
S10 = (10/2) (a1 + a10) = 1650000
So 5(a1 + a10) = 1650000
or a1 + a10 = 330000.
a1 + 9d = 33000
and this is another equation relating a1 and d. So we have two simultaneous linear equations for a1 and d.

Solve with standard simultaneous equation methods.
-a1 + 12d = 0
a1 + 9d = 330000
For instance, elimination. If you add those two equations, the a1 is eliminated, you can solve for d, and then plug that back into either equation to get a1.

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