Let `f(x)=1+x^(111)+x^(222)+x^(333)…………….+x^(999)` then `f(x)` is divisible by

Options

A. `x+1`
B. x
C. `x-1`
D. `1+x^(222)+x^(444)+x^(666)+x^(888)`

Answer

To solve the problem, we need to analyze the function \( f(x) = 1 + x^{111} + x^{222} + x^{333} + \ldots + x^{999} \).

### Step 1: Recognize the Pattern
The function \( f(x) \) consists of a series of terms where the exponents are multiples of 111, starting from \( 0 \) (which is \( 1 \)) up to \( 999 \). The exponents can be expressed as \( 111n \) where \( n \) ranges from \( 0 \) to \( 9 \).

### Step 2: Write the Function in Summation Form
We can express \( f(x) \) as:
\[ f(x) = \sum_{n=0}^{9} x^{111n} \]

### Step 3: Identify the Number of Terms
The number of terms in this summation is \( 10 \) (from \( n=0 \) to \( n=9 \)).

### Step 4: Use the Formula for the Sum of a Geometric Series
The sum of a geometric series can be calculated using the formula:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

In our case:
- \( a = 1 \)
- \( r = x^{111} \)
- \( n = 10 \)

Thus, we can write:
\[ f(x) = \frac{1 - (x^{111})^{10}}{1 - x^{111}} = \frac{1 - x^{1110}}{1 - x^{111}} \]

### Step 5: Analyze the Divisibility
From the expression \( f(x) = \frac{1 - x^{1110}}{1 - x^{111}} \), we can see that \( f(x) \) is a polynomial in \( x \). The numerator \( 1 - x^{1110} \) indicates that \( f(x) \) will be divisible by \( 1 - x^{111} \) because it is a factor of the numerator.

### Conclusion
Thus, \( f(x) \) is divisible by \( 1 - x^{111} \).