The 1st,3nd and 9th term of an AP are the first three term of a GP.if the 7th term of the AP is 14. Calculate the 20th term of the AP and sum of the first twelve term of the GP?

Let the first term of the AP be a and the common difference be d.

Given that the 1st, 3rd, and 9th terms of the AP are the first three terms of a GP, we have:

a, a + 2d, a + 8d

Since these are the first three terms of a GP, we have:

(a + 2d)^2 = (a)(a + 8d)

Expanding and simplifying, we get:

a^2 + 4ad + 4d^2 = a^2 + 8ad

4ad + 4d^2 = 8ad

4d^2 = 4ad

d = a

Therefore, the common difference of the AP is equal to the first term.

Given that the 7th term of the AP is 14, we have:

a + 6a = 14

7a = 14

a = 2

Therefore, the first term of the AP is 2 and the common difference is also 2.

The 20th term of the AP can be calculated as:

a + 19d = 2 + 19(2) = 40

Therefore, the 20th term of the AP is 40.

The sum of the first twelve terms of the GP can be calculated using the formula:

S = a(r^n - 1) / (r - 1)

Where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.

Since the first three terms of the GP are a, a + 2d, a + 8d, the common ratio is (a + 8d) / (a + 2d) = (2 + 8(2)) / (2 + 2(2)) = 18 / 6 = 3

Therefore, the sum of the first twelve terms of the GP is:

S = 2(3^12 - 1) / (3 - 1) = 2(531441 - 1) / 2 = 531440

Therefore, the sum of the first twelve terms of the GP is 531440.

Given:
- 1st, 3rd, and 9th terms of AP are also the first three terms of a GP.
- 7th term of AP = 14

Let's assume the AP is:
a, a + d, a + 2d, a + 3d, ...

And the GP is:
ar, ar², ar³

From the given information:

- a = ar (1st term of AP and GP)
- a + 2d = ar² (3rd term of AP and GP)
- a + 8d = ar³ (9th term of AP and GP)

Equation 2:
a + 2d = ar²
Dividing by a, we get:
1 + 2d/a = r²

Equation 3:
a + 8d = ar³
Dividing by a, we get:
1 + 8d/a = r³

Equation 4:
7th term of AP = a + 6d = 14

To find the 20th term of AP:
a + 19d = a + 19(2) [Since d = 2]
= a + 38

To find the sum of the first twelve terms of GP:
S = ar(1 - r^12) / (1 - r)

We don't have the value of r directly. However, we can use Equation 1 and Equation 2 to find r.

From Equation 1:
a = ar
Cancelling 'a', we get:
1 = r

Substituting r = 1 in Equation 2, we get:
1 + 2d/a = 1²
=> 2d/a = 0
=> d = 0

Therefore, the AP is an arithmetic progression with common difference 0. This means it is a constant progression where all terms are equal to the first term.

Hence, the 20th term of AP is:
a + 19d = a + 0 = a

And the sum of the first twelve terms of GP is:
S = ar(1 - r^12) / (1 - r) = ar(1 - 1^12) / (1 - 1) = 0