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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 $$\pi = \frac{22}{7}$$ )

We know,
Perimeter of a semi-circle = $$\pi$$r

Therefore,
$$P_1 = \pi(0.5) = {{\pi} \over {2}}$$ cm
$$P_2 = \pi(1) = \pi$$ cm
$$P_3 = \pi(1.5) = {{3pi} \over { 2}}$$ cm

Where, $$P_1, P_2, P_3$$ are the lengths of the semi-circles.

Hence we got a series here, as,

$${{\pi} \over {2}} , \pi, {{3\pi} \over { 2}}, 2\pi,$$ ….

$$P_1 = {{\pi} \over {2}}$$ cm
$$P_2 = \pi$$ cm
Common difference, $$d = P_2- P_1 = \pi – {{\pi} \over {2}} = {{\pi} \over {2}}$$
First term = $$P_1= a = {{\pi} \over {2}}$$ cm

By the sum of n term formula, we know,
$$S_n = {{n} \over {2}} [2a + (n – 1)d]$$

Therefor, Sum of the length of 13 consecutive circles is;

$$S_{13} = {{13} \over {2}} [2({{\pi} \over {2}}) + (13 – 1){{\pi} \over {2}}]$$
= $$\frac{13}{2} [\pi + 6\pi]$$
=$$\frac{13}{2} (7\pi)$$
= $$\frac{13}{2} × 7 × \frac{22}{7}$$
= $$143$$ cm