Question

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.

Answer 1

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.

Answer 2

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.