The problem is to find a formula that we can use to 
fill in the empty blanks of this sequence:

___,___,___,___,___,___,-10,___,___,___,___,___,-48


Let's assume it's an arithmetic sequence.  Then we use

an = a1 + (n-1)d

Substitute n=7

an = a1 + (7-1)d
a7 = a1 + 6d

when n=7, an = -10, so a7 = -10

Substituting a7 = -10

-10 = a1 + 6d

Also,

an = a1 + (n-1)d

Substitute n=13

a13 = a1 + (13-1)d
a13 = a1 + 12d

when n=13, an = -48, so a13 = -48

Substituting a13 = -48

-48 = a1 + 12d

So we have the system of two equations and
two unknowns

-10 = a1 + 6d
-48 = a1 + 12d

So we solve that system and get:

a1 = 28, d = -19/3

So to get the formula, we take

an = a1 + (n-1)d

And substitute those values of a1 and d

an = 

an = 

an = 

an = 

That's the formula.  We can now fill in the blanks:

28,65/3,46/3,9,8/3,-11/3,-10,-49/3,-68/3,-29,-106/3,-48.

-------------------------------------
That's the solution your teacher wants, most likely.

However, it is also possible that it could be a 
geometric sequence as well:

Let's assume it's a geometric sequence.  Then we use

an = a1rn-1

Substitute n=7

a7 = a1r7-1
a7 = a1r6

when n=7, an = -10, so a7 = -10

Substituting a7 = -10

-10 = a1r6

Also,

an = a1rn-1

Substitute n=13

a13 = a1r13-1
a13 = a1r12

when n=13, an = -48, so a13 = -48

Substituting a13 = -48

-48 = a1r12

So we have the system of two equations and
two unknowns

-10 = a1r6
-48 = a1r12

So we solve that system and get:

a1 = , d = 

So to get the formula, we take

an = a1rn-1

And substitute those values of a1 and d

an = 

That's the formula.  We can now fill in the blanks,
but I'll just give their decimal values:

-2.083,-2.706,-3.514,-4.564,-5.928,-7.699,
-10,-12.99,-16.87,-21.91,-28.46,-36.96,-48.

Actually, arithmetic and geometric sequences are only 2 kinds of
many sequences.  So technically there are infinitely many answers to
this problem, but only one arithmetic sequence solution, and only one
geometric sequence solution. 

Edwin