THE 1ST, 3RD AND 9TH TERM OF A LINEAR SEQUENCE A.P ARE THE 1ST THREE TIMES OF G.P. IF THE 7TH TERM OF THE LINEAR SEQUENCE IS 14. CALCULATE (A) 20TH TERM OF THE LINEAR SEQUENCE (B) SUM OF THE 1ST, 12TH TERM OF THE G.P

If 14th terms of an A.P is twice 8th term.If 6th term is -8,than find the sum of its first 20 term

To solve this problem, we need to find the common difference of the arithmetic progression (A.P) and the common ratio of the geometric progression (G.P).

Let's denote the first term of the A.P as 'a', and the common difference as 'd'.
Similarly, let's denote the first term of the G.P as 'b', and the common ratio as 'r'.

Given that the 1st, 3rd, and 9th terms of the A.P are the 1st, 2nd, and 3rd terms of the G.P, we can set up the following equations:

1st term of A.P = 1st term of G.P: a = b
3rd term of A.P = 2nd term of G.P: a + 2d = br
9th term of A.P = 3rd term of G.P: a + 8d = br^2

We are also given that the 7th term of the A.P is 14:
7th term of A.P: a + 6d = 14

Now, let's solve these equations.

From the equation a + 2d = br, we can express 'a' in terms of 'b' and 'r':
a = br - 2d

Substituting this value of 'a' in the equation a + 8d = br^2, we get:
(br - 2d) + 8d = br^2
br + 6d = br^2

Rearranging terms, we have:
br^2 - br - 6d = 0

This is a quadratic equation in terms of 'r'. We need to solve this equation to find the value of 'r'. We can do this by using the quadratic formula:

r = (-b ± √(b^2 - 4ac)) / (2a)

In our equation, a = 1, b = -1, and c = -6d.

Plugging in these values, we get:
r = (1 ± √(1^2 - 4(-6d))) / (2)

Simplifying further, we have:
r = (1 ± √(1 + 24d)) / 2

Since r represents the common ratio of the G.P, we know that r is positive. Therefore, taking the positive value, we get:
r = (1 + √(1 + 24d)) / 2

Now that we have the value of 'r' in terms of 'd', we can substitute this in the equation a + 6d = 14 to solve for 'd':

a + 6d = 14
(b + 4d) + 6d = 14 (substituting a = b)
b + 10d = 14

We can express 'b' in terms of 'd':
b = a = br - 2d
b = (1 + √(1 + 24d)) / 2

Substituting this value in the equation, we get:
((1 + √(1 + 24d)) / 2) + 10d = 14

Now, we can solve this equation to find the value of 'd'.

a+2d=14