# Thread: Problem Involving Arithmetic progression

1. ## Problem Involving Arithmetic progression

Diagram shows the waves formed when a stone is thrown into the water.each wave has the shape of a circle and is $\displaystyle r cm$ away from the one before it.

a)show that the circumferences of the waves form an arithmetic progression and find the common difference.

b)If the radius if the smallest wave is $\displaystyle 4cm$ , find the radius of the tenth wave.Hence,find its circumference in term of $\displaystyle \pi$

2. Originally Posted by mastermin346
Diagram shows the waves formed when a stone is thrown into the water.each wave has the shape of a circle and is $\displaystyle r cm$ away from the one before it.

a)show that the circumferences of the waves form an arithmetic progression and find the common difference.

b)If the radius if the smallest wave is $\displaystyle 4cm$ , find the radius of the tenth wave.Hence,find its circumference in term of $\displaystyle \pi$

Hi mastermin346,

(a)

The radius of the inner circle is r cm. $\displaystyle C=2\pi r$
The radius of the middle circle is 2r cm. $\displaystyle C=4 \pi r$
The radius of the outer circle is 3r cm. $\displaystyle C=6 \pi r$

The circumferences form an arithmetic progression with a common difference of $\displaystyle 2 \pi r$

(b)

The radius of the inner circle is 4 cm. $\displaystyle C=8\pi$
The radius of the middle circle is 8 cm. $\displaystyle C=16 \pi$
The radius of the outer circle is 12 cm. $\displaystyle C=24 \pi$

The common difference in the sequence of circumferences $\displaystyle 8\pi,\: 16\pi,\: 24\pi, \:. . .$ is $\displaystyle 8\pi$

The common difference in the radius sequence of 4, 8, 12,..... is 4

To find the radius of the 10th circle use $\displaystyle r_{10}=r_1+(n-1)d$

$\displaystyle r_{10}=4+(10-1)4$

$\displaystyle r_{10}=40$

Use this radius of the 10th circle to compute the circumference: $\displaystyle C=\pi D$

$\displaystyle C=80 \pi$