Answer to Question #263820 in Algebra for Syazwina Roslan

Question

 A boy is swinging on playground exercise gym. He swings through an arc of 1 m on his first swing. On each successive swing, he swings through an arc 70% of the previous length.

a) Show that the total distance the boy swings in his first n swings is 10

given by S  3 1 0.7 where n 1,2,3,.... nn

Hence, find the total distance travelled by the boy after six swings.

Answer 1

Let $a_n$  denote the distance the boy swings in the  $n^{th}$ swing.

Then, according to the given information, the sequence $a_n$  is a geometric sequence with first term  $a_n=1$ and common ratio $r=70\%=0.7$

Hence, the general term is given by $a_n.r^{n-1}=1.07^{n-1}=0.7^{n-1}$

Consequently, the total distance the boy swings in his first n swings is as follows:

$S_n=a_1+a_2+a_3.......+a_n\\=1+0.7^1+0.7^2+0.7^3+........+0.7^{n-1}\\=0.7s_n=0.7+0.7^2+0.7^3+......+0.7^n\\s_n-0.7 s_n=(1+0.7^1+0.7^2+0.7^3+......+0.7^{n-1}-(0.7+0.7^2+0.7^3+....+0.7^n)\\=0.3s_n=1-0.7^n\\=\frac{3}{10}s_n=1-0.7^n\\s_n=\frac{10}{3}(1-0.7^n)$

Therefore, the total distance the boy swings in his first  n swings is given by $S_n=\frac{10}{3}(1-0.7^n)$  where $n=1,2,3$

Further,  $S_6=\frac{10}{3}(1-0.7^6)=2.94$

Therefore, the total distance travelled by the boy after six swings is approximately 2.94 m.