Solve the equation: -4 + (-1) + 2 +……+ x = 437

To solve the equation:

-4 + (-1) + 2 + ... + x = 437,

we observe that it forms an arithmetic sequence. Let's break it down step by step.

Step 1: Identify the pattern
The terms in the sequence alternate between negative and positive integers. The first few terms are:

-4, -1, 2, ...

Notice that each term increases by 3 from the previous term. So, this is an arithmetic sequence with:

First term (a) = -4
Common difference (d) = 3
Step 2: General formula for the nth term
The formula for the nth term of an arithmetic sequence is:

Tn = a + (n - 1) * d

Here,

a = -4
d = 3
So the nth term of the sequence is:

Tn = -4 + (n - 1) * 3 = -4 + 3n - 3 = 3n - 7

Step 3: Find the sum of the sequence
The sum of the first n terms of an arithmetic sequence is given by:

S_n = n/2 * (first term + last term)

We know the sum is 437, so we can set up the equation:

437 = n/2 * (-4 + x)

Where x is the last term (which is the nth term, so x = Tn).

Now, substitute the expression for the nth term:

437 = n/2 * (-4 + (3n - 7))

Simplify the equation:

437 = n/2 * (3n - 11)

Multiply both sides by 2 to eliminate the fraction:

874 = n * (3n - 11)

Distribute n:

874 = 3n^2 - 11n

Step 4: Solve the quadratic equation
Rearrange the terms to form a standard quadratic equation:

3n^2 - 11n - 874 = 0

Now, we solve this quadratic equation using the quadratic formula:

n = [-(-11) ± √((-11)^2 - 4 * 3 * (-874))] / (2 * 3)

Simplify:

n = [11 ± √(121 + 10488)] / 6 n = [11 ± √10609] / 6 n = [11 ± 103] / 6

Now, calculate the two possible values for n:

n = (11 + 103) / 6 = 114 / 6 = 19 n = (11 - 103) / 6 = -92 / 6 = -15.33 (This value is not valid, as n must be a positive integer)

Thus, n = 19.

Step 5: Find the value of x
Now that we know n = 19, we can find the 19th term (x). Using the formula for the nth term:

x = 3n - 7 x = 3(19) - 7 x = 57 - 7 x = 50

Final Answer:
The value of x is 50.