Paulo makes a sequence of numbers.
he chooses a starting number and then subtracts equal amounts each time.
the third number in his sequence is 45.
the tenth number is -32.
what is the first number in his sequence.
show your method.

45 - -32 = 45 + 32 = 77.  
10 - 3 = 7.
77 /7 = 11. 
Paulo takes 11 away from each number. 

11 + 11 = 22. 
45 + 22 = 67. 
The first number was 67. 

The sequence is an Arithmetic Progression.
T  =  a + (n-1)d.
Where a = first term.
           d = common difference.
           n =  Number of term

let the first number = a.

Sequence =  a,  a -d, a-2d, a-3d,

3rd =  a -2d = 45      (i)
10th  =  a -9d = -32  (ii)

 (i) minus (ii).

(a -2d) - (a -9d) = 45 - (-32)
a -2d -a +9d  = 77
-2d + 9d  = 77
7d  = 77.  Divide by 7.
d = 77/7 = 11.

Substitute d =11, in (i)
a-2d = 45.
a - 2(11) = 45
a -22 = 45.
a = 45 + 22 = 67,
Therefore first term = 67.

There should only be 1 solution