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 Bird Hobbs. in South Africa
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

.

Solution

Let’s break down the problem step by step.

Part 1: Quadratic Sequence

The sequence given is .
  1. Identify the differences:
    • First difference:
    • Second difference:
    • Third difference:
    • Fourth difference:
  2. Calculate the first differences:
    • First difference:
    • Second difference:
    • Third difference:
    • Fourth difference:
  3. Set up the equations based on the differences:
    • Since the second differences are constant in a quadratic sequence, we have: Simplifying gives:
  4. Now find :
    • Using the same logic: Simplifying gives:
Thus, and .

Part 2: Series Calculation

The series is .

2.2.1 Calculate the sum of the series.

  1. Identify the first term and the common difference :
    • First term
    • Common difference
  2. Find the number of terms :
    • The last term is .
    • Using the formula for the -th term of an arithmetic series: Setting :
  3. Calculate the sum :

2.2.2 Write the series in sigma-notation.

The series can be expressed in sigma notation as:

Part 3: Bouncing Ball

A ball is dropped from a height of m and loses of its height on each bounce.
  1. Calculate the total distance:
    • The height after each bounce is .
    • The total distance is:
    • The series is a geometric series with first term and ratio :
    • Thus, the total distance:
The total distance the ball will bounce cannot exceed m.

Part 4: Series Convergence

Given:

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

The series converges if the common ratio :
This leads to:

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

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

Part 5: Sum of the First Terms

Given:
To find the term :
  1. The sum of the first terms is given by:
  2. The -th term can be expressed as:
  3. We need to find and :
  4. Thus:
The term is (

Answered by UpStudy AI and reviewed by a Professional Tutor

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Simplify this solution

The Deep Dive

In a quadratic sequence, each term is generated by adding an increasing amount, known as the second difference, to the previous term. Here, we can see that the difference between 6 and , as well as the subsequent differences, will lead us to find and by establishing the quadratic relationship among the terms.
For the series , it’s an arithmetic sequence where the first term is 220 and the common difference is -7. To find the sum, we can use the formula for the sum of an arithmetic series. The series has 34 terms, and its sum is .

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