6. Find 4 arithmetic means between 5 and 20.
(a) Write the formula to calculate arithmetic mean between a and b.
(b) What is the 4th mean of the given sequence? Find it.
(c) Show that the second middle number is 11.
Answer (1)
The arithmetic mean between a and b is 2 a + b .
Calculate the common difference: d = 5 20 − 5 = 3 .
Find the 4 t h arithmetic mean: a 4 = 5 + 4 × 3 = 17 .
Verify the second arithmetic mean: a 2 = 5 + 2 × 3 = 11 . The final answer is 17 .
Explanation
Understanding the Problem We are given two numbers, 5 and 20, and we need to insert 4 arithmetic means between them. This means we will have an arithmetic sequence with 6 terms in total. We need to find the formula for the arithmetic mean between two numbers, calculate the value of the 4th arithmetic mean, and show that the 2nd arithmetic mean is 11.
Arithmetic Mean Formula (a) The arithmetic mean between two numbers a and b is simply their average, which is given by the formula: 2 a + b This formula provides the value that lies exactly in the middle of a and b .
Finding the 4th Arithmetic Mean (b) Let the arithmetic sequence be 5 , a 1 , a 2 , a 3 , a 4 , 20 . Here, a 1 , a 2 , a 3 , and a 4 are the four arithmetic means we want to find. The first term of the sequence is a = 5 , and the last term is l = 20 . The number of terms in the sequence is n = 6 . We can find the common difference d using the formula for the last term of an arithmetic sequence: l = a + ( n − 1 ) d . Substituting the given values, we get: 20 = 5 + ( 6 − 1 ) d 20 = 5 + 5 d 15 = 5 d d = 3 Now that we have the common difference d = 3 , we can find the 4 t h arithmetic mean, a 4 , using the formula a 4 = a + 4 d . Substituting the values, we get: a 4 = 5 + 4 ( 3 ) = 5 + 12 = 17 So, the 4 t h arithmetic mean is 17.
Showing the 2nd Arithmetic Mean is 11 (c) To show that the second arithmetic mean is 11, we can use the formula a 2 = a + 2 d . Substituting the values a = 5 and d = 3 , we get: a 2 = 5 + 2 ( 3 ) = 5 + 6 = 11 Thus, the second arithmetic mean is indeed 11.
Conclusion In summary, the arithmetic mean between a and b is 2 a + b . The 4 t h arithmetic mean of the given sequence is 17, and the second arithmetic mean is 11, as required.
Examples
Arithmetic sequences are useful in many real-life situations. For example, imagine you are saving money each month, increasing the amount you save by a fixed amount. If you start by saving $5 in the first month and increase your savings by $3 each month, the amounts you save each month form an arithmetic sequence: $5, $8, $11, $14, and so on. This problem demonstrates how to calculate specific terms in such a sequence, helping you predict how much you'll save in a particular month.
