The sequence `9, 18, 27, 36, 45, 54,…..` consists of successive mutiple of 9. This sequence is then altered by multiplying every other term by `-1,` starting with the first term, to prduce the new sequence `-9, 18, -27,36,-45, 54,……..` If the sum of the first n terms of this new sequence is 180, determine n.

Answer

To solve the problem, we need to find the value of \( n \) such that the sum of the first \( n \) terms of the new sequence equals 180. The new sequence is formed by alternating the signs of the terms in the original sequence of multiples of 9.

### Step-by-Step Solution:

1. **Identify the Original Sequence:**
The original sequence is:
\[ 9, 18, 27, 36, 45, 54, \ldots \]
This can be expressed as:
\[ a_r = 9r \quad (r = 1, 2, 3, \ldots) \]

2. **Alter the Sequence:**
The new sequence is formed by multiplying every other term by -1, starting with the first term:
\[ -9, 18, -27, 36, -45, 54, \ldots \]
The general term of the new sequence can be expressed as:
\[ t_r = (-1)^{r+1} \cdot 9r \]

3. **Sum of the First \( n \) Terms:**
We need to find the sum \( S_n \) of the first \( n \) terms:
\[ S_n = \sum_{r=1}^{n} t_r = \sum_{r=1}^{n} (-1)^{r+1} \cdot 9r \]
Factoring out the 9:
\[ S_n = 9 \sum_{r=1}^{n} (-1)^{r+1} r \]

4. **Calculate the Sum \( \sum_{r=1}^{n} (-1)^{r+1} r \):**
The sum can be simplified based on whether \( n \) is even or odd:
- If \( n \) is even, say \( n = 2k \):
\[ \sum_{r=1}^{2k} (-1)^{r+1} r = (1 - 2) + (3 - 4) + \ldots + (2k-1 - 2k) = -k \]
- If \( n \) is odd, say \( n = 2k + 1 \):
\[ \sum_{r=1}^{2k+1} (-1)^{r+1} r = (1 - 2) + (3 - 4) + \ldots + (2k - 2k) + (2k + 1) = k + 1 \]

5. **Set Up the Equation:**
We know \( S_n = 180 \):
\[ 9 \sum_{r=1}^{n} (-1)^{r+1} r = 180 \]
This simplifies to:
\[ \sum_{r=1}^{n} (-1)^{r+1} r = 20 \]

6. **Consider Cases for \( n \):**
- **Case 1:** \( n = 2k \) (even)
\[ -k = 20 \implies k = -20 \quad \text{(not possible)} \]
- **Case 2:** \( n = 2k + 1 \) (odd)
\[ k + 1 = 20 \implies k = 19 \implies n = 2(19) + 1 = 39 \]

7. **Conclusion:**
The value of \( n \) that satisfies the condition is:
\[ n = 39 \]