The eleventh term of an A.P.with positive terms is four times its third term and the product of thord te and eleventh term is 256.Find the A.P.

thord te ?????

"The eleventh term of an A.P.with positive terms is four times its third term"
---> a+11d = 4(a+2d)
a+11d = 4a + 6d
5d = 3a
d = 3a/5

"the product of thord te and eleventh term is 256"
I will read that as third term

(a+2d)(a+11d) = 256
a^2 + 13ad + 11d^2 = 256
a^2 + 13a(3a/5) + 11(9a^2/25) = 256
a^2 + 39a^2/5 + 99a^2/25 = 256
times 25
25a^2 + 180a^2 + 99a^2 = 6400
304a^2 = 6400
a^2 = 6400/304 = 400/19
a = ± 20/√19
d = ±(3/5)(20/√19) = ±12/√19

AP is 20/√19 , 32/√19, 44/√19, ...

once you fix your typo, make any necessary changes to my solution

11th term is a+10d

that will make a difference.

To find the arithmetic progression (A.P.), we need to find its first term (a) and common difference (d).

Given information:
1) The eleventh term (a₁₁) is four times its third term (a₃).
2) The product of the third term (a₃) and the eleventh term (a₁₁) is 256.

Let's solve for the unknowns step by step:

Step 1: Express the terms in terms of the first term (a) and common difference (d).
a₁₁ = a + (11-1)d = a + 10d (Equation 1)
a₃ = a + (3-1)d = a + 2d (Equation 2)

Step 2: Use the given information to form equations and solve them simultaneously.
From the first given information, we have:
a₁₁ = 4(a₃)
Equalizing the two expressions using Equation 1 and Equation 2, we get:
a + 10d = 4(a + 2d)
a + 10d = 4a + 8d
3a = 2d (Equation 3)

From the second given information, we have:
a₃ * a₁₁ = 256
Substituting the values from Equation 1 and Equation 2, we get:
(a + 2d) * (a + 10d) = 256
Expanding and simplifying, we get:
a² + 12ad + 20d² = 256 (Equation 4)

Step 3: Solve the simultaneous equations (Equation 3 and Equation 4) to find the values of a and d.
From Equation 3, we have:
3a = 2d
a = (2/3)d

Substituting this value of a in Equation 4, we get:
(a² + 12ad + 20d²) = 256
((2/3)d)² + 12((2/3)d)d + 20d² = 256
(4/9)d² + 8d² + 20d² = 256
(32/9)d² + 20d² = 256
(32d² + 180d²)/9 = 256
212d²/9 = 256
212d² = 2304
d² = 2304/212
d² = 10.8868
d ≈ ±√10.8868
d ≈ ± 3.3 (rounded to one decimal place)

Step 4: Substitute the value of d to find the value of a.
Taking d = 3.3, the value of a can be found from Equation 3:
3a = 2d
3a = 2 * 3.3
3a = 6.6
a ≈ 2.2 (rounded to one decimal place)

Taking d = -3.3, the value of a can be found from Equation 3:
3a = 2d
3a = 2 * (-3.3)
3a = -6.6
a ≈ -2.2 (rounded to one decimal place)

Therefore, the two possible arithmetic progressions (A.P.) are:
1) A.P. with first term (a) ≈ 2.2 and common difference (d) ≈ 3.3
2) A.P. with first term (a) ≈ -2.2 and common difference (d) ≈ -3.3