Question

2- suppose a ball is thrown upward to a height of h0 meters . each time the ball bounces, it rebounds to a fraction r of its previous height . let hn be the hight after the nth term biunces. cosider the following value of h0 and r

a- find the first 4 terms of the sequence of heights(hn)

b- find a general expression for the nth term of the sequence (hn)

h0=20 , r=0.5

Answer 1

Answer:

a) 40, 50, 55, 57.5

b) $S_n=h_0+2h_0\sum_{n=1}^{\infty}r^n$

Step-by-step explanation:

h₀ = Initial height of the ball =20

r = Rebound fraction = 0.5

a) The series of bouncing balls is given by

Sₙ=h₀+2h₀(r¹+r²+r³+r⁴.........rⁿ)

S₁=h₀+2h₀r¹=20+2×20×0.5=40

S₂=h₀+2h₀(r¹+r²)=20+2×20×(0.5+0.5²)=50

S₃=h₀+2h₀(r¹+r²+r³)=20+2×20×(0.5+0.5²+0.5³)=55

S₄=h₀+2h₀(r¹+r²+r³+r⁴)=20+2×20×(0.5+0.5²+0.5³+0.5⁴)=57.5

b) General expression for the nth term of the sequence

$S_n=h_0+2h_0\sum_{n=1}^{\infty}r^n$