Question

The vertical lines in the figure below are drawn 1 centimetres apart.

Prove that their heights are in arithmetic sequence.

What is the common difference?

Answer 1

Given: BC = CD = DE = 1 cm

Let AB = x cm

∴ AC = (x + 1) cm, AD = (x + 2) cm, AE = (x + 3) cm

In ΔABF, ∠ABF = 90°

In ΔACG, ∠ACG = 90°

In ΔADH, ∠ADH = 90°

In ΔAEI, ∠AEI = 90°

The sequence of the heights is , + 0.8391, + 2(0.8391), + 2(0.8391), …

Here, first term =

Second term = + 0.8391

Third term = + 2(0.8391)

Second term − First term = + 0.8391 − = 0.8391

Third term − Second term = + 2(0.8391) − − 0.8391 = 0.8391

Since, third term − second term = second term − first term, the sequence of heights is an arithmetic sequence with common difference 0.8391.