The solution of the equation `1+4+7+ . . . . . .. +x=925` is :

Options

A. 73
B. 76
C. 70
D. 74

Answer

To solve the equation \(1 + 4 + 7 + \ldots + x = 925\), we can follow these steps:

### Step 1: Identify the series
The series \(1, 4, 7, \ldots\) is an arithmetic progression (AP) where:
- The first term \(a = 1\)
- The common difference \(d = 4 - 1 = 3\)

### Step 2: Find the number of terms
Let \(n\) be the number of terms in the series. The \(n\)-th term of an AP can be expressed as:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the values of \(a\) and \(d\):
\[ a_n = 1 + (n - 1) \cdot 3 = 1 + 3n - 3 = 3n - 2 \]
Thus, the last term \(x\) can be expressed as:
\[ x = 3n - 2 \]

### Step 3: Use the formula for the sum of an AP
The sum \(S_n\) of the first \(n\) terms of an AP is given by:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]
Substituting the known values:
\[ S_n = \frac{n}{2} \cdot (2 \cdot 1 + (n - 1) \cdot 3) = \frac{n}{2} \cdot (2 + 3n - 3) = \frac{n}{2} \cdot (3n - 1) \]
Setting this equal to 925:
\[ \frac{n}{2} \cdot (3n - 1) = 925 \]

### Step 4: Solve for \(n\)
Multiply both sides by 2:
\[ n(3n - 1) = 1850 \]
Expanding this gives:
\[ 3n^2 - n - 1850 = 0 \]

### Step 5: Use the quadratic formula
The quadratic formula is:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 3\), \(b = -1\), and \(c = -1850\). Plugging in these values:
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 3 \cdot (-1850)}}{2 \cdot 3} \]
Calculating the discriminant:
\[ b^2 - 4ac = 1 + 22200 = 22201 \]
Now substituting back into the formula:
\[ n = \frac{1 \pm \sqrt{22201}}{6} \]

### Step 6: Calculate \(\sqrt{22201}\)
Calculating the square root gives approximately \(149\):
\[ n = \frac{1 \pm 149}{6} \]
This results in two potential solutions:
1. \(n = \frac{150}{6} = 25\)
2. \(n = \frac{-148}{6}\) (not valid since \(n\) must be positive)

### Step 7: Find the value of \(x\)
Now, substituting \(n = 25\) back to find \(x\):
\[ x = 3n - 2 = 3 \cdot 25 - 2 = 75 - 2 = 73 \]

### Final Answer
Thus, the solution of the equation \(1 + 4 + 7 + \ldots + x = 925\) is:
\[ \boxed{73} \]