How i solve this semi circles question of arithmetic progressions of ncert of exercise 5.3 of class 10th math, please suggest me the best way to solve this problem A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take π = 22/7)
We know,
Perimeter of a semi-circle = πr
Therefore,
P1 = π(0.5) = π/2 cm
P2 = π(1) = π cm
P3 = π(1.5) = 3π/2 cm
Where, P1, P2, P3 are the lengths of the semi-circles.
Hence we got a series here, as,
π/2, π, 3π/2, 2π, ….
P1 = π/2 cm
P2 = π cm
Common difference, d = P2 – P1 = π – π/2 = π/2
First term = P1= a = π/2 cm
By the sum of n term formula, we know,
Sn = n/2 [2a + (n – 1)d]
Therefor, Sum of the length of 13 consecutive circles is;
S13 = 13/2 [2(π/2) + (13 – 1)π/2]
= 13/2 [π + 6π]
=13/2 (7π)
= 13/2 × 7 × 22/7
= 143 cm