The radii of n circular discs, labelled 1,2,...,n, from smallest to largest, follow an arithmetic progression.
The radius of disc 4 is 11 units. The sum of the radii of the first 4 discs is 26 units.
Disc 2 to disc n are coloured with (n-1) different colours such that disc k is coloured with colour k, for k = 2,3,...,n. The n discs are then stacked in decreasing order of radii with disc n at the bottom and disc 1 on top such that a vertical line passes through the centres of all the n discs. The figure below shows a stack with 4 discs when viewed from the top.
It is given that Uk, k = 2,...,n, represents the area of region with colour k in the stack with n discs when viewed from the top.
(i) Show that
(a) the radius of the disc k is (3k-1) units,
(b) Uk = 3π (6k -5 )
(ii) Show that the sequence U2, U3, U4, ..., Un is an arithmetic progression.
*************************
Answer:
(i)
(a)
Disc 4 = a + 3d = 11 units -------- (1)
Sum of radii of first 4 discs = 2 (2a + 3d)
= 4a + 6d = 26 units --------- (2)
(1) × 2: 2a + 6d = 22 ------ (3)
(2) - (3): 2a = 4
a = 2
thus, 2 + 3d = 11
3d = 9
d = 3
Thus, radius of disc k = 2 + (k-1) × 3
= 2 + 3k - 3
= (3k - 1) units (shown)
(b)
Hence, radius of disc k-1 = (3(k-1) - 1)
= 3k - 4
Uk = π(3k - 1)² - π(3k - 4)²
= π (9k² - 6k + 1) - π (9k² - 24k + 16)
= π (18k -15)
= 3π (6k - 5) (shown)
(ii)
Uk = 18π k - 15π
Uk-1 = 18π (k-1) - 15π = 18π k - 33π
Uk - Uk-1 = 18π = constant
Hence, the sequence U2, U3, U4, ..., Un is an arithmetic progression (shown)
