6. There are 4 arithmetic means between 5 and 20.
(a) Write the formula to calculate arithmetic mean between [tex]$a$[/tex] and [tex]$b$[/tex].
(b) What is the 4th mean of the given sequence? Find it.
(c) Show that the second middle number is 11.
Answer (2)
The formula for the arithmetic mean between a and b is 2 a + b .
The common difference d is calculated as 3.
The four arithmetic means are 8, 11, 14, and 17, so the 4th arithmetic mean is 17.
The second arithmetic mean is indeed 11, as shown by the calculations. 17
Explanation
Understanding the Problem Let's break down this problem step by step. We're given two numbers, 5 and 20, and we need to insert four arithmetic means between them. This means we'll have a sequence of six numbers in total, where the difference between consecutive terms is constant. We need to find the formula for the arithmetic mean between two numbers, calculate the value of the fourth arithmetic mean, and verify that the second arithmetic mean is 11.
Finding the Formula for Arithmetic Mean (a) The arithmetic mean between two numbers a and b is simply their average. So, the formula is: 2 a + b
Calculating the Arithmetic Means (b) Now, let's find the four arithmetic means between 5 and 20. Let these means be A 1 , A 2 , A 3 , and A 4 . This gives us the arithmetic sequence: 5, A 1 , A 2 , A 3 , A 4 , 20. Let d be the common difference between consecutive terms. Since there are 6 terms in the sequence, we can write the last term (20) in terms of the first term (5) and the common difference d as follows: 20 = 5 + ( 6 − 1 ) d Simplifying this equation, we get: 20 = 5 + 5 d Subtracting 5 from both sides: 15 = 5 d Dividing by 5: d = 3 Now we can find the arithmetic means: A 1 = 5 + 3 = 8 A 2 = 8 + 3 = 11 A 3 = 11 + 3 = 14 A 4 = 14 + 3 = 17 So, the four arithmetic means are 8, 11, 14, and 17. The fourth arithmetic mean, A 4 , is 17.
Verifying the Second Arithmetic Mean (c) From our calculations in part (b), we found that the second arithmetic mean, A 2 , is indeed 11. This confirms that the second middle number is 11.
Examples
Arithmetic sequences and means are useful in various real-life scenarios, such as calculating loan payments or determining the spacing of elements in a design. For instance, if you want to install equally spaced shelves between two fixed points, you can use arithmetic means to determine the position of each shelf. Suppose you want to place 4 shelves between a height of 5 feet and 20 feet. Using the arithmetic sequence we calculated, the shelves would be placed at heights of 8, 11, 14, and 17 feet, ensuring equal spacing and a visually appealing design.
The formula for the arithmetic mean between two numbers is 2 a + b . The four arithmetic means between 5 and 20 are 8, 11, 14, and 17, with the 4th mean being 17. The second mean is confirmed to be 11, as calculated.
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