Question

Suppose a ball is thrown upward to a height of h 0 meters. Each time the ball​ bounces, it 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. 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) first 4 terms would be 10, 5, 2.5, 1.25.

b) $h_n=20(0.5)^{n-1}$

Step-by-step explanation:

Since we have given that

h₀ = 20

r = 0.5

As it is a geometric sequence.

So, hₙ would be

$h_n=h_0r^{n-1}\\\\h_n=20(0.5)^{n-1}$

First 4 terms would be

$h_1=20(0.5)=10\\\\h_2=20(0.5)^2=5\\\\h_3=20(0.5)^3=2.5\\\\h_4=20(0.5)^4=1.25$

Hence, a) first 4 terms would be 10, 5, 2.5, 1.25.

b) $h_n=20(0.5)^{n-1}$

Answer 2

The terms of the sequence and the general expression of the sequence were gotten as;

A)1st term:45

2nd term:48.75

3rd term:49.6875

4th term:49.921875

B) Sₙ = h₀ + 2h₀((∞, n=1) Σrⁿ)

What is an arithmetic sequence?

We are given;

h₀ = Initial height of the ball = 20

r = Rebound fraction = 0.5

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 = 20 + (2 × 20 × 0.5)

S₁ = 40

The second term of the sequence is;

S₂ = h₀ + 2h₀(r¹+r²)

S₂ = 20 + (2 × 20 × (0.5 + 0.5²)) = 50

The third term of the sequence is;

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

S₄ = h₀ + 2h₀(r¹ + r² + r³ + r⁴)

S₄ = 20 + (2 × 20 × (0.5 + 0.5² + 0.5³ + 0.5⁴)) = 57.5

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

Sₙ = h₀ + 2h₀((∞, n=1) Σrⁿ)

Read more about Arithmetic Sequence at; https://brainly.com/question/7882626