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.
Since the second differences are constant in a quadratic sequence, we have:Simplifying gives:
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.
Identify the first term and the common difference :
First term
Common difference
Find the number of terms :
The last term is .
Using the formula for the -th term of an arithmetic series:Setting :
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.
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 :
The sum of the first terms is given by:
The -th term can be expressed as:
We need to find and :
Thus:
The term is (
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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 .