Find the arithmetic progression and sum of its first 20 terms whose arithmetic mean and geometric mean between first and third term is 10 and 8 respectively
To find the arithmetic progression and sum of its first 20 terms, we need to first determine the common difference (d) and the first term (a1) of the progression.
Let's assume the first term of the arithmetic progression as 'a'. Since the arithmetic mean of the first, second, and third term is given as 10, we can write the following equation:
(a + a + d)/3 = 10
Simplifying this expression, we get:
2a + d = 30 --- Equation (1)
We are also given that the geometric mean between the first and third term is 8. The geometric mean between two numbers is the square root of their product. Therefore, we can write the following equation:
sqrt(a(a + 2d)) = 8
Simplifying this expression, we get:
√(a^2 + 2ad) = 8
Squaring both sides, we have:
a^2 + 2ad = 64 --- Equation (2)
Now we have a system of two equations (Equation 1 and Equation 2) with two unknowns (a and d). We can solve these equations simultaneously to find the values of 'a' and 'd'.
To solve this system of equations, let's multiply Equation 1 by 2:
4a + 2d = 60 --- Equation (3)
Subtract Equation 2 from Equation 3:
(4a + 2d) - (a^2 + 2ad) = 60 - 64
Rearranging the terms, we get:
a^2 - 2a - 4d = -4
Now, we have a quadratic equation. We can solve this equation to find the value of 'a'.
a^2 - 2a - 4d + 4 = 0
We can factorize this equation:
(a - 4)(a + 1) = 0
From this, we get two possible values for 'a': a = 4 or a = -1.
Let's consider a = 4.
Substituting 'a' = 4 in either Equation 1 or Equation 2, we can find the value of 'd'. Let's substitute it in Equation 1:
2(4) + d = 30
8 + d = 30
d = 30 - 8
d = 22
Therefore, we have found the common difference (d = 22) and the first term (a = 4) of the arithmetic progression.
Now, we can find the terms of the progression and calculate the sum of the first 20 terms.
The terms of the arithmetic progression are given by:
an = a1 + (n - 1)d
Substituting the values of a1 = 4, d = 22, and n = 1 to 20 into this formula, we can find all 20 terms of the progression.
First term (a1) = 4
Second term (a2) = 4 + 22 = 26
Third term (a3) = 4 + 2(22) = 48
...
Continue this process until you find the 20th term.
Finally, to calculate the sum of the first 20 terms, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a1 + an)
Substituting the values of n = 20, a1 = 4, and an (20th term found above), we can find the sum of the first 20 terms of the arithmetic progression.