Solve the equation `(x+1)+(x+4)+(x+7)++(x+28)=155.`

Explanation

To solve the equation \((x+1)+(x+4)+(x+7)+\ldots+(x+28)=155\), we will follow these steps:

### Step 1: Identify the sequence
The terms in the equation can be expressed as:
- \(x + 1\)
- \(x + 4\)
- \(x + 7\)
- ...
- \(x + 28\)

This is an arithmetic progression (AP) where the first term \(a = x + 1\) and the common difference \(d = 3\).

### Step 2: Find the number of terms
To find the number of terms, we note that the last term is \(x + 28\). We can use the formula for the \(n\)-th term of an AP:
\[ a_n = a + (n-1)d \]
Setting \(a_n = x + 28\), we have:
\[ x + 28 = (x + 1) + (n - 1) \cdot 3 \]
This simplifies to:
\[ 28 = 1 + (n - 1) \cdot 3 \]
\[ 27 = (n - 1) \cdot 3 \]
\[ n - 1 = 9 \]
\[ n = 10 \]

### Step 3: Calculate the sum of the series
The sum \(S_n\) of the first \(n\) terms of an AP is given by:
\[ S_n = \frac{n}{2} \cdot (a + a_n) \]
Substituting \(n = 10\), \(a = x + 1\), and \(a_n = x + 28\):
\[ S_{10} = \frac{10}{2} \cdot ((x + 1) + (x + 28)) \]
\[ S_{10} = 5 \cdot (2x + 29) \]

### Step 4: Set the sum equal to 155
We set the sum equal to 155:
\[ 5 \cdot (2x + 29) = 155 \]

### Step 5: Solve for \(x\)
Dividing both sides by 5:
\[ 2x + 29 = 31 \]
Subtracting 29 from both sides:
\[ 2x = 31 - 29 \]
\[ 2x = 2 \]
Dividing by 2:
\[ x = 1 \]

### Final Answer
The value of \(x\) is \(1\).

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