Solve the equation for x:

1 + 4 + 7 + 10 + ………. + x = 287

To solve the equation \(1 + 4 + 7 + 10 + \ldots + x = 287\), we first need to recognize that the series on the left side is an arithmetic sequence. An arithmetic sequence is defined by a starting number, a common difference, and a number of terms. In this case, the first term is 1, and the common difference is 3 (since each term increases by 3). Let's break this down step by step.

Identifying the Sequence

The sequence can be expressed as:

• First term (a) = 1
• Common difference (d) = 3

The general formula for the nth term of an arithmetic sequence is given by:

a_n = a + (n - 1)d

Substituting the values we have:

a_n = 1 + (n - 1) * 3 = 3n - 2

Finding the Number of Terms

Next, we need to determine how many terms are in the sequence up to \(x\). We can express the sum of the first \(n\) terms of an arithmetic sequence using the formula:

S_n = n/2 * (a + a_n)

Here, \(S_n\) is the sum of the first \(n\) terms, \(a\) is the first term, and \(a_n\) is the nth term. We know that:

S_n = 287

Substituting the known values into the formula gives us:

287 = n/2 * (1 + (3n - 2))

This simplifies to:

287 = n/2 * (3n - 1)

Multiplying both sides by 2 to eliminate the fraction:

574 = n(3n - 1)

Expanding this gives:

3n² - n - 574 = 0

Solving the Quadratic Equation

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

n = (-b ± √(b² - 4ac)) / 2a

In our equation, \(a = 3\), \(b = -1\), and \(c = -574\). Plugging these values into the formula:

n = (1 ± √((-1)² - 4 * 3 * (-574))) / (2 * 3)

This simplifies to:

n = (1 ± √(1 + 6888)) / 6

n = (1 ± √6889) / 6

Calculating the square root gives us:

√6889 = 83

Thus, we have:

n = (1 ± 83) / 6

This results in two possible solutions:

• n = (84) / 6 = 14
• n = (-82) / 6 (not valid since n must be positive)

Finding the Value of x

Since we found \(n = 14\), we can now find \(x\) using the nth term formula:

x = a_n = 3n - 2

Substituting \(n = 14\):

x = 3(14) - 2 = 42 - 2 = 40

Final Result

Therefore, the value of \(x\) that satisfies the equation \(1 + 4 + 7 + 10 + \ldots + x = 287\) is:

x = 40