Question

Suppose a ball is thrown upward to a height of h 0 meters. After each​ bounce, the ball rebounds to a fraction r of its previous height. Let h Subscript n be the height after the nth bounce. Consider the following values of h 0 and r. Complete parts​ (a) and​ (b) below. h0=30​, r=0.25

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)

Answer 1

A)1st term:45

2nd term:48.75

3rd term:49.6875

4th term:49.921875

B) Sₙ = h₀ + 2h₀((∞, n=1) Σrⁿ)

Explanation:

We are given;

h₀ = Initial height of the ball = 30

r = Rebound fraction = 0.25

a) The arithmetic sequence of bouncing balls is given by the following;

Sₙ=h₀+2h₀(r¹+r²+r³+r⁴.........rⁿ)

The first term of the sequence is;

S₁ = h₀ + 2h₀r¹

S₁ = 30 + (2 × 30 × 0.25)

S₁ = 45

The second term of the sequence is;

S₂ = h₀ + 2h₀(r¹+r²)

S₂ = 30 + (2 × 30 × (0.25 + 0.25²)) = 48.75

The third term of the sequence is;

S₃ = h₀ + 2h₀(r¹ + r² + r³) = 30 + (2 × 30 × (0.25 + 0.25² + 0.25³)) = 49.6875

S₄ = h₀ + 2h₀(r¹ + r² + r³ + r⁴)

S₄ = 30 + (2 × 30 × (0.25 + 0.25² + 0.25³ + 0.25⁴)) = 49.921875

B) The general expression for the nth term of the sequence is;

Sₙ = h₀ + 2h₀((∞, n=1) Σrⁿ)