Each term of a progression is determined by adding 0.5 to the preceding term. the sum of the first 25 terms of the progression equals the square of the 25th term. calculate the possible value(s) of the first term

As usual, let the first term be a, and we know d = .5

S25 = 25/2[2a + 24(.5)] = 25a + 150

T25 = a + 24d = a + 12

given: S25 = (T25)^2

25a + 150 = (a+12)^2
a^2 + 24a + 144 = 25a + 150
a^2 - a - 6 = 0
(a-3)(a+2) = 0

a = 3 or a = -2

To solve this problem, let's first identify what we know:

1. The progression follows a pattern in which each term is determined by adding 0.5 to the preceding term.
2. The sum of the first 25 terms of the progression equals the square of the 25th term.

Let's break down the problem step by step:

Step 1: Find the formula for the nth term of the progression.
In this case, the formula for the nth term can be written as: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the position of the term in the sequence, and d is the common difference.

Since the common difference is 0.5, the formula becomes: an = a1 + (n - 1) * 0.5.

Step 2: Calculate the sum of the first 25 terms of the progression.
The sum of an arithmetic series can be found using the formula: Sn = (n/2)(2a1 + (n - 1)d), where Sn is the sum of the first n terms.

In this case, the formula becomes: S25 = (25/2)(2a1 + (25 - 1) * 0.5).

Step 3: Express the sum as the square of the 25th term.
According to the problem, the sum of the first 25 terms of the progression equals the square of the 25th term. Therefore, we can write:

(25/2)(2a1 + (25 - 1) * 0.5) = (a1 + (25 - 1) * 0.5)^2.

Step 4: Simplify the equation and solve for a1.
Let's expand and simplify the equation to solve for a1:

(25/2)(2a1 + 12) = (a1 + 12)^2.

Divide both sides of the equation by 25 to simplify:

(2a1 + 12) = (a1 + 12)^2 / 25.

Now, we have a quadratic equation:

(2a1 + 12) = (a1^2 + 24a1 + 144) / 25.

Multiply both sides by 25 to remove the denominator:

50a1 + 300 = a1^2 + 24a1 + 144.

Rearrange the equation to match the quadratic form:

0 = a1^2 - 26a1 - 156.

Step 5: Solve the quadratic equation.
To solve the quadratic equation, we can factor it or use the quadratic formula. In this case, let's use the quadratic formula:

a1 = (-b +/- sqrt(b^2 - 4ac)) / 2a.

For the equation a1^2 - 26a1 - 156 = 0, the coefficients are: a = 1, b = -26, c = -156.

Using the quadratic formula, we find:

a1 = (-(-26) +/- sqrt((-26)^2 - 4(1)(-156))) / 2(1),
a1 = (26 +/- sqrt(676 + 624)) / 2,
a1 = (26 +/- sqrt(1300)) / 2,
a1 = (26 +/- 36.06) / 2.

Simplifying further, we have two possible solutions for a1:

a1 = (26 + 36.06) / 2 ≈ 31.03,
a1 = (26 - 36.06) / 2 ≈ -5.03.

Therefore, the possible values for the first term of the progression are approximately 31.03 and -5.03.