The sum of the 3 terms in arithmetic sequence is 39.if these numbers is increased by 1,5 and 12 respectively the numbers would be in geometric progression. find the second term of the geometric progression?

3/2 (2a+2d) = 39

(a+d+5)/(a+1) = (a+2d+12)/(a+d+5)

a=11, d=2
AP: 11,13,15
GP: 12,18,27

a=26, d= -13
AP: 26, 13, 0
GP: 27,18,12

Why did the arithmetic sequence enroll in a geometry class? Because it wanted to learn about geometric progressions! Now, let's solve this mathemagical problem together.

Let's represent the first term of the arithmetic sequence as "a" and the common difference as "d". So, our three terms would be "a", "a + d", and "a + 2d".

According to the problem, the sum of these three terms is 39, so we can write the equation: a + (a + d) + (a + 2d) = 39.

Simplifying this equation, we get: 3a + 3d = 39.

Dividing both sides of the equation by 3, we find: a + d = 13. (Equation 1)

Now, let's look at the geometric progression formed by increasing each number by 1, 5, and 12. This means the terms of the geometric progression would be: a + 1, a + d + 5, and a + 2d + 12.

Since it's a geometric progression, we know that the ratio between these terms would be the same. We can set up an equation using this information:

(a + d + 5) / (a + 1) = (a + 2d + 12) / (a + d + 5).

Simplifying this equation, we get: (a + d + 5)² = (a + 2d + 12)(a + 1).

Expanding both sides, we get: a² + 3ad + 4d² + 10a + 30d + 25 = a² + 3ad + 2a + 2d + 12a + 24d + 12.

Canceling out similar terms, we have: 4d² + 10a + 30d + 25 = 2a + 2d + 12a + 24d + 12.

Bringing everything to one side, we get: 4d² + 25a + 6d - 14 = 0. (Equation 2)

Now, we have two equations—Equation 1 and Equation 2—with two unknowns, "a" and "d". Let's solve them simultaneously.

Subtracting Equation 1 from Equation 2, we get: 25a + 6d - 14 - (a + d) = 0.

Simplifying this equation, we find: 24a + 5d - 14 = 0.

Hmm, this looks a bit tricky to solve. I think I might need a clown calculator to find the correct arithmetic solution.

To solve this problem, let's call the first term of the arithmetic sequence "a" and the common difference "d".

The three terms can be represented as:
First term: a
Second term: a + d
Third term: a + 2d

According to the given information, these three terms form a geometric progression when increased by 1, 5, and 12 respectively. This means:

(a + 1) / a = (a + d + 5) / (a + d) = (a + 2d + 12) / (a + 2d)

Let's solve these equations step-by-step:

Step 1: Simplify the first equation:
(a + 1) / a = (a + d + 5) / (a + d)
(a + 1)(a + d) = a(a + d + 5)
(a^2 + ad + a + ad + d^2 + 5d) = a^2 + ad + 5a
a^2 + 2ad + d^2 + 5d = a^2 + ad + 5a

Step 2: Simplify and rearrange the equation:
ad + d^2 + 5d = ad + 5a
d^2 + 5d = 5a

Step 3: Rearrange the equation:
d^2 + 5d - 5a = 0

Step 4: Solve the quadratic equation:
Using the quadratic formula, d = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = 5, and c = -5a.

d = (-5 ± √(5^2 - 4(1)(-5a))) / (2(1))
d = (-5 ± √(25 + 20a)) / 2

Step 5: Substitute the possible values of "a" to find the corresponding values of "d":
Since we don't know the exact value of "a" yet, we will substitute different values to find the solution.

For example, if we substitute a = 1:
d = (-5 ± √(25 + 20(1))) / 2
d = (-5 ± √(45)) / 2

If we substitute a = 2:
d = (-5 ± √(25 + 20(2))) / 2
d = (-5 ± √(65)) / 2

Step 6: Test the values of "d" in the original equation:
Using the values obtained in Step 5, substitute "d" back into the original equation (a + 2d + 12) / (a + 2d) = (a + 1) / a and solve for "a".

For each value of "d", solve for "a". Remember that the sum of the arithmetic sequence is 39. Once you find the values of "a" and "d", calculate the second term of the geometric progression, which is a + d.

To find the second term of the geometric progression, we need to follow a step-by-step process. Let's break it down:

Step 1: Find the arithmetic mean (average) of the three numbers from the arithmetic progression.
Since we know that the sum of the three terms in the arithmetic sequence is 39, we divide that sum by 3 to find the arithmetic mean:
Arithmetic mean = Sum of terms / Number of terms
Arithmetic mean = 39 / 3
Arithmetic mean = 13

Step 2: Increase each term of the arithmetic sequence by the given values (1, 5, and 12, respectively). We now have three numbers that form a geometric progression.

Step 3: Determine the common ratio of the geometric progression.
To find the common ratio, we divide the third term by the second term (as all increased values form a geometric progression):
Common ratio = third term / second term

Since the third term is obtained by increasing the arithmetic mean by 12, and the second term is obtained by increasing the arithmetic mean by 5, we have:
Common ratio = (Arithmetic mean + 12) / (Arithmetic mean + 5)

Now, substitute the value of the arithmetic mean (which we found to be 13) into the equation:
Common ratio = (13 + 12) / (13 + 5)
Common ratio = 25 / 18

Step 4: Find the second term of the geometric progression.
To find the second term, we multiply the first term by the common ratio.
Since the first term is obtained by increasing the arithmetic mean by 1, we have:
Second term = (Arithmetic mean + 1) * Common ratio

Substituting the value of the arithmetic mean and the common ratio:
Second term = (13 + 1) * (25 / 18)
Second term = 14 * (25 / 18)
Second term = 350 / 18
Second term ≈ 19.44