Consider the series $$ (2 x+2)+(3 x+3)+(4 x+4)+(5 x+5)+\ldots+(30 x+30) $$, (1) Determine the number of terms in the series. (2) Calculate the sum of the series in terms of $$ x $$, (3) For which value of $$ n $$ is $$ S_{n}=2555 x+2555 $$ ?

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Let's analyze the series $$ (2x + 2) + (3x + 3) + (4x + 4) + \ldots + (30x + 30) $$ step by step.

### 1. **Number of Terms**

The series starts with $$2x + 2$$ and ends with $$30x + 30$$. Each term increases both the coefficient of $$x$$ and the constant term by 1.

- **First term:** $$k = 2$$
- **Last term:** $$k = 30$$

The number of terms is calculated as:
$$
\text{Number of terms} = 30 - 2 + 1 = 29
$$

### 2. **Sum of the Series in Terms of $$x$$**

Each term of the series can be expressed as $$kx + k$$, where $$k$$ ranges from 2 to 30.

The sum of the series $$S$$ is:
$$
S = \sum_{k=2}^{30} (kx + k)
$$
$$
S = x \sum_{k=2}^{30} k + \sum_{k=2}^{30} k
$$
$$
S = (x + 1) \sum_{k=2}^{30} k
$$

First, compute the sum of integers from 2 to 30:
$$
\sum_{k=2}^{30} k = \sum_{k=1}^{30} k - 1 = \frac{30 \times 31}{2} - 1 = 465 - 1 = 464
$$

Therefore, the sum of the series is:
$$
S = (x + 1) \times 464 = 464x + 464
$$

### 3. **Determining $$n$$ for $$S_n = 2555x + 2555$$**

Assume $$S_n$$ represents the sum of the first $$n$$ terms of a similar series starting from $$2x + 2$$.

The general form of the sum is:
$$
S_n = (x + 1) \sum_{k=2}^{n} k = (x + 1) \left( \frac{n(n + 1)}{2} - 1 \right)
$$
$$
S_n = \left( \frac{n(n + 1)}{2} - 1 \right)x + \left( \frac{n(n + 1)}{2} - 1 \right)
$$

Set $$S_n$$ equal to $$2555x + 2555$$ and equate the coefficients:
$$
\frac{n(n + 1)}{2} - 1 = 2555
$$
$$
\frac{n(n + 1)}{2} = 2556
$$
$$
n(n + 1) = 5112
$$

Solving the quadratic equation:
$$
n^2 + n - 5112 = 0
$$
$$
n = \frac{-1 \pm \sqrt{1 + 4 \times 5112}}{2} = \frac{-1 \pm 143}{2}
$$

Taking the positive root:
$$
n = \frac{142}{2} = 71
$$

### **Final Answers**

1. **Number of terms:** $$29$$
2. **Sum of the series:** $$464x + 464$$
3. **Value of $$n$$:** $$71$$

Answer:
1. There are 29 terms;
2. The sum is 464 × x plus 464;
3. The value of n is 71.
1. There are 29 terms. 2. The sum of the series is $$464x + 464$$. 3. The value of $$n$$ is 71.

Consider the series \( (2 x+2)+(3 x+3)+(4 x+4)+(5 x+5)+\ldots+(30 x+30) \), (1) Determine the number of terms in the series. (2) Calculate the sum of the series in terms of \( x \), (3) For which value of \( n \) is \( S_{n}=2555 x+2555 \) ?

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