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)

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# 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) Q.18

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We know,

Perimeter of a semi-circle = π

rTherefore,_{P}_{1}= π(0.5) = π/2 cm_{P}_{2}= π(1) = π cm_{P}_{3}= π(1.5) = 3π/2 cmWhere, P

_{1,}P_{2},P_{3}are the lengths of the semi-circles.Hence we got a series here, as,

π/2, π, 3π/2, 2π, ….

P= π/2 cm

_{1}P_{2}= π cmCommon difference, d=P_{2 }– P_{1}= π – π/2 = π/2First term = P

_{1}=a= π/2 cmBy the sum of n term formula, we know,

S=_{n}n/2 [2a+ (n– 1)d]Therefor, Sum of the length of 13 consecutive circles is;

S=_{13}13/2 [2(π/2) + (13 – 1)π/2]=

13/2 [π + 6π]=13/2 (7π)=

13/2 × 7 ×22/7= 143 cm