The second term, the fifth term and eleventh term are in a Gp.if the eleventh term is 4.Find the difference between the sums of each progression

better read what you posted.

difference between the sums of each progression ??

you have only described a single progression: the gp.

And how can there be a difference between each? A difference is between two things. Bad grammar, vague description, sloppy math.

Clean it up and we'll be happy to help find an answer. Don't make us figure out the question as well!

To solve this problem, we need to find the common ratio of the geometric progression (GP) and then calculate the sums of the first 11 terms for two different series. The difference between the sums will give us the result. Here are the steps to get the answer:

Step 1: Find the common ratio (r) of the GP.
Since the second, fifth, and eleventh terms are in a GP, we can use the formula for the nth term of a GP:
Tₙ = a * r^(n-1)

Setting up two equations using the given information:
T₂ = a * r^(2-1)
T₅ = a * r^(5-1)

Since T₂ and T₅ are given, we can write the equations as:
a * r = T₂
a * r^4 = T₅

Step 2: Solve the equations to find the values of a and r.
Divide the second equation by the first equation to eliminate a:
(a * r^4) / (a * r) = T₅ / T₂
r^3 = T₅ / T₂

Substituting the values given:
r^3 = 4 / T₂

Since the eleventh term is given as 4:
r^3 = 4 / T₂

Step 3: Calculate the values of a and r.
Using the formula Tₙ = a * r^(n-1), we can substitute n=11 and Tₙ=4:
4 = a * r^(11-1)
4 = a * r^10

Step 4: Calculate the sums of the first 11 terms for two different series.
First series: Sum of the first 11 terms of the GP with the common ratio r:
S₁₁ = a * (r^11 - 1) / (r - 1)

Second series: Sum of the first 11 terms of the GP with the common ratio (1/r) to cancel out the terms:
S'₁₁ = a * (1 - (1/r)^11) / (1 - (1/r))

Step 5: Find the difference between the sums of each progression.
Difference = S₁₁ - S'₁₁

Note: To get the exact numerical difference, you will need to substitute the values of T₂ and T₅ into the equations and perform the calculations.