Arithmetic Progression – Finding values of unknown elements in a sequence

If 5, x, y, 14 … is an arithmetic progression, find the values of x and y

Suggested solution

Let us recall in an arithmetic progression, AP, the common difference is usually constant between terms.

For this, we deduce as follows

14 - y = y – x = x – 5

If 3 things are equal to one another in Mathematics, eg,

a = b = c,

It follows that

a = b,  a = c, and b = c

From the above, simultaneous equations van be formed as follows

If 14 - y = y – x = x – 5, then

14 – y = y – x      (we mark this equ1)

14 – y = x – 5      (we mark this equ2)

From equ1, we make x the subject as follows

14 – y = y – x

14 – y – y = –x

14 – 2y = –x

x = 2y – 14          (we call this equ3)

Substitute 2y-14 for x in equ2 as follows

14 – y = x – 5

14 – y = 2y - 14 – 5

14 + 14 + 5 = 2y + y

14 + 14 + 5 = 2y + y

33 = 3y

y = ³³/₃

y = 11

Substitute 11 for y in equ3 as follows

x = 2y – 14

x = 2(11) – 14

x = 22 – 14

x = 8