Insert five arithmetic means between 7 and 17.
so you have 7 terms with a=7 and
a + 6d = 17
7 + 6d = 17
d = 10/6 = 5/3
form the missing terms,
e.g term(2) = 7 + 5/3 = 26/3
term(3) = 26/3 + 5/3 = 31/3
etc
To insert five arithmetic means between 7 and 17, we need to find the common difference between each consecutive term.
The common difference for an arithmetic sequence can be calculated using the formula:
Common difference (d) = (final term - first term) / (number of terms - 1)
In this case, the first term (a1) is 7, the final term (an) is 17, and the number of terms (n) is 7 (including the given terms).
d = (17 - 7) / (7 - 1)
d = 10 / 6
d = 5/3
Now that we have the common difference (d), we can insert the arithmetic means between 7 and 17 by adding multiples of the common difference to the first term.
The terms for the arithmetic sequence are:
7, (7 + 5/3), (7 + 10/3), (7 + 15/3), (7 + 20/3), (7 + 25/3), 17
Simplifying those terms:
7, (7 + 5/3), (7 + 10/3), (7 + 15/3), (7 + 20/3), (7 + 25/3), 17
7, (7 + 1.67), (7 + 3.33), (7 + 5), (7 + 6.67), (7 + 8.33), 17
7, 8.67, 10.33, 12, 13.67, 15.33, 17
So, the arithmetic means between 7 and 17 are 8.67, 10.33, 12, 13.67, and 15.33.
To insert five arithmetic means between two numbers, you need to find the common difference between the terms. In this case, the common difference will be the difference between each term and the previous term.
To find the common difference, subtract the first term from the second term:
17 - 7 = 10
Now, divide the common difference by the number of means plus one (since we want to insert five means between the two terms):
Common difference = 10 / (5 + 1) = 10 / 6 = 1.6667 (rounded to four decimal places)
Now, starting from the first term (7), add the common difference to get the sequence of arithmetic means:
7 + 1.6667 = 8.6667
8.6667 + 1.6667 = 10.3333
10.3333 + 1.6667 = 12.0000
12.0000 + 1.6667 = 13.6667
13.6667 + 1.6667 = 15.3333
15.3333 + 1.6667 = 17.0000
Therefore, the sequence of arithmetic means between 7 and 17 is:
7, 8.6667, 10.3333, 12.0000, 13.6667, 15.3333, 17