**CBSE NCERT Solutions Chapter 5 Arithmetic Progressions Exercise 5.3 Question 18**

**18. A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ... What is the total length of such a spiral made up of thirteen consecutive semicircles.**

**Solution:**

Length of semi-circle = Circumference of circle/2 = =

Length of semi circle of radii 0.5 cm = cm

Length of semi circle of radii 1.0 cm = cm

Length of semi circle of radii 1.5 cm = cm

**Therefore, we have sequence of the form:**

...... 13 terms {There are total of thirteen semi-circles}.

To find total length of the spiral, we need to find sum of the sequence ...... 13 terms

Total length of spiral ...... 13 terms

Total length of spiral = .... 13 terms ** (1)**

Sequence 0.5, 1.0, 1.5 ....13 terms is an arithmetic progression.

Lets find the sum of this sequence.

First term = a= 0.5

Common difference = 1.0 - 0.5 = 0.5

n = 13

**Applying formula**, to find sum of n terms of AP ,** we get**

Therefore, 0.5 + 1.0 + 1.5 + 2.0 ....13 terms = 45.5

Putting this in equation **(1), **we get

Total length of spiral = ) terms = = 143 cm