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.

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.

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.