Question
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ESTION 2
Consider the following quadratic sequence: Determine the values of and .
2 Given the following series:
2.2.1 Calculate the sum of the series.
2.2.2 Write the series in sigma-notation.
2.3 A batt is dropped from a beight of 15 m . It bounces back and loses of its prev height on each bounce. Show that the total distance the ball will bounce cannot exi 290 m .
2.4 Given: .
2.4.1 For which value(s) of will the series converge?
2.4.2 If , calculate the sum to infinity of the series if it exists.
2.5 The sum of the first terms of a sequence is .
Deternine the term. Leave your answer in the form where and at integers.

Ask by Lee Bradley. in South Africa
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

  • The sum of the series is 3553.
  • The series in sigma-notation is .
  • The total distance the ball will bounce is 285 m.
  • The series converges for . When , the series does not converge.
  • The 70th term is .

Solution

Let’s break down the problem step by step.

1. Quadratic Sequence:

A quadratic sequence has a constant second difference. Let’s denote the terms as follows:
First, we calculate the first differences:
Now, the first differences are:
Next, we calculate the second differences:
Since the second differences must be constant, we set them equal:
Let’s solve these equations.

Solving the first equation:

Solving the second equation:

Thus, the values are:

2. Series:

This is an arithmetic series where:
  • First term
  • Common difference

2.1 Calculate the sum of the series.

To find the number of terms , we use the formula for the -th term of an arithmetic sequence:
Setting :
Now, we can calculate the sum of the first terms:

2.2 Write the series in sigma-notation.

The series can be expressed in sigma notation as:

3. Bouncing Ball Problem

A ball is dropped from a height of m and loses of its height on each bounce. The height after each bounce forms a geometric series.
The heights are:
  • First drop:
  • First bounce:
  • Second bounce:
  • And so on…
The total distance is:
The series is a geometric series with:
  • First term
  • Common ratio
The sum of an infinite geometric series is given by:
Thus, the total distance is:

4. Series Convergence

Given:
This series converges if the common ratio . The common ratio is .

4.1 For which value(s) of will the series converge?

Multiplying by :
This gives:

4.2 If , calculate the sum to infinity of the series if it exists.

Since is outside the convergence range, the series does not converge.

5. Sum of the First Terms

Given:
To find the term, we need to express the -th term :
Assuming :
Thus,
For ( n =

Answered by UpStudy AI and reviewed by a Professional Tutor

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Bonus Knowledge

The quadratic sequence you presented can be approached by recognizing that the second difference will remain constant. By determining the sequence’s first differences and then the second difference, you can find the missing values and . If you observe the relationship, you’ll find and .
As for the series , it’s an arithmetic series with a first term of 220, a common difference of -7, and a last term of -11. You can calculate the number of terms using the formula for the nth term of an arithmetic sequence. The sum can be computed using the formula , where is the first term, and is the last term. With some calculations, you’ll find that the total sum is 1,323. As for sigma-notation, this series can be expressed as .

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