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 step by step, we can follow these instructions:

### Step 1: Identify the new sequence
The original sequence is:
\[ 9, 18, 27, 36, 45, 54, \ldots \]
After altering the sequence by multiplying every other term by -1, the new sequence becomes:
\[ -9, 18, -27, 36, -45, 54, \ldots \]

### Step 2: Write the general term of the new sequence
The general term \( T_r \) of the new sequence can be expressed as:
\[ T_r = (-1)^{r} \cdot 9r \]
where \( r \) is the term number (1, 2, 3, ...).

### Step 3: Set up the sum of the first n terms
The sum of the first n terms \( S_n \) of the new sequence is given by:
\[ S_n = \sum_{r=1}^{n} T_r = \sum_{r=1}^{n} (-1)^{r} \cdot 9r \]
This can be factored as:
\[ S_n = 9 \sum_{r=1}^{n} (-1)^{r} r \]

### Step 4: Set the sum equal to 180
We know from the problem statement that:
\[ S_n = 180 \]
Thus, we have:
\[ 9 \sum_{r=1}^{n} (-1)^{r} r = 180 \]
Dividing both sides by 9 gives:
\[ \sum_{r=1}^{n} (-1)^{r} r = 20 \]

### Step 5: Analyze the alternating sum
The sum \( \sum_{r=1}^{n} (-1)^{r} r \) can be computed based on whether \( n \) is even or odd:
- If \( n \) is even, say \( n = 2k \):
\[ \sum_{r=1}^{2k} (-1)^{r} r = (-1)(1) + (2) + (-3)(3) + (4) + \ldots + (-1)(2k-1) + (2k) \]
This results in:
\[ = k \]
- If \( n \) is odd, say \( n = 2k + 1 \):
\[ \sum_{r=1}^{2k+1} (-1)^{r} r = (-1)(1) + (2) + (-3)(3) + (4) + \ldots + (-1)(2k-1) + (2k) + (-(2k+1)) \]
This results in:
\[ = - (k + 1) \]

### Step 6: Solve for n
Since we need the sum to equal 20, we can only have \( n \) as even:
\[ k = 20 \implies n = 2k = 40 \]

### Final Answer
Thus, the value of \( n \) is:
\[ n = 40 \]