Write three arithmetic and three geometric means between 4 and 324
So in each case you would have 5 terms
AS:
a = 4, t(5) = a+4d = 324
4 + 4d = 324
4d = 320,
d = 80
the three missing terms are 84 , 164, 244
GS:
a = 4, t(5) = ar^4 = 324
r^4 = 81
r = ± 3
if r = 3, the three missing terms would be
12, 36, 108
if r = -3 , the three missing terms would be
-12 , +36, -108
find the 6th term 4 and 324
To find the arithmetic mean between two numbers, we calculate the average by adding them together and dividing by 2.
Arithmetic Mean #1:
(4 + 324) / 2 = 328 / 2 = 164
Arithmetic Mean #2:
(4 + 164) / 2 = 168 / 2 = 84
Arithmetic Mean #3:
(4 + 84) / 2 = 88 / 2 = 44
To find the geometric mean between two numbers, we take the square root of their product.
Geometric Mean #1:
√(4 * 324) = √(1296) = 36
Geometric Mean #2:
√(4 * 36) = √(144) = 12
Geometric Mean #3:
√(4 * 12) = √(48) ≈ 6.928
Arithmetic Mean:
To find arithmetic means, we need to first identify the common difference between the given terms.
The terms we have are 4 and 324, and we want to find 3 arithmetic means between them.
The common difference (d) can be calculated using the formula:
d = (last term - first term) / (number of terms + 1)
In this case, the first term is 4, the last term is 324, and the number of terms is 5 (2 given terms + 3 arithmetic means).
So, we have:
d = (324 - 4) / (5 + 1)
d = 320 / 6
d = 53.3333333333 (rounded to 10 decimal places)
Now, we can find the arithmetic means by adding the common difference to the previous terms:
First arithmetic mean:
4 + 53.3333333333 = 57.3333333333
Second arithmetic mean:
57.3333333333 + 53.3333333333 = 110.6666666666
Third arithmetic mean:
110.6666666666 + 53.3333333333 = 164
Geometric Mean:
To find geometric means, we need to take the square root of the product of the given terms and the desired geometric means.
The terms we have are 4 and 324, and we want to find 3 geometric means between them.
The geometric mean (G) can be calculated using the formula:
G = √(first term * last term) ^ (1 / number of terms + 1)
In this case, the first term is 4, the last term is 324, and the number of terms is 5 (2 given terms + 3 geometric means).
So, we have:
G = √(4 * 324) ^ (1/6)
G = √(1296) ^ (1/6)
G = √(1296) ^ 0.1666666666 (rounded to 10 decimal places)
Now, we can find the geometric means by taking the (1/6)th power of the product:
First geometric mean:
(4 * 324) ^ 0.1666666666 = 8.2540404996
Second geometric mean:
(8.2540404996 * 324) ^ 0.1666666666 = 15.1902755671
Third geometric mean:
(15.1902755671 * 324) ^ 0.1666666666 = 27.9511453825
Therefore, the three arithmetic means between 4 and 324 are 57.3333333333, 110.6666666666, and 164.
The three geometric means between 4 and 324 are 8.2540404996, 15.1902755671, and 27.9511453825.