What is the value of `1+i^(2) +i^(4) + i^(6) +"………"i^(100)`, where `i=sqrt(-1)`

Options

A. 0
B. 1
C. `-1`
D. None of these

Answer

To solve the expression \(1 + i^2 + i^4 + i^6 + \ldots + i^{100}\), we will follow these steps:

### Step 1: Identify the pattern of powers of \(i\)
The powers of \(i\) (where \(i = \sqrt{-1}\)) cycle every four terms:
- \(i^0 = 1\)
- \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\) (and the cycle repeats)

### Step 2: Determine the values of even powers of \(i\)
Since we are only interested in even powers of \(i\), we can calculate:
- \(i^2 = -1\)
- \(i^4 = 1\)
- \(i^6 = -1\)
- \(i^8 = 1\)
- Continuing this pattern, we see that:
- For even powers \(i^{2n}\):
- If \(n\) is odd, \(i^{2n} = -1\)
- If \(n\) is even, \(i^{2n} = 1\)

### Step 3: Identify the range of \(n\)
The series goes from \(i^2\) to \(i^{100}\). The even powers from \(2\) to \(100\) can be represented as \(i^{2n}\) where \(n\) ranges from \(1\) to \(50\) (since \(2n\) must be even and \(n\) can take values from \(1\) to \(50\)).

### Step 4: Count the contributions from the series
- The even indices \(n\) (i.e., \(2, 4, 6, \ldots, 100\)) correspond to:
- \(n = 1, 3, 5, \ldots, 49\) (which are odd \(n\)) contribute \(-1\) each.
- \(n = 2, 4, 6, \ldots, 50\) (which are even \(n\)) contribute \(1\) each.

### Step 5: Calculate the total contributions
- There are \(50\) terms in total.
- The odd indexed terms (which are \(1, 3, 5, \ldots, 49\)) amount to \(25\) terms contributing \(-1\) each.
- The even indexed terms (which are \(2, 4, 6, \ldots, 50\)) also amount to \(25\) terms contributing \(1\) each.

### Step 6: Sum the contributions
The total contribution from the odd indexed terms is:
\[ 25 \times (-1) = -25 \]
The total contribution from the even indexed terms is:
\[ 25 \times 1 = 25 \]
Now, adding these contributions together:
\[ -25 + 25 = 0 \]

### Step 7: Add the initial term
Now, we need to add the initial term \(1\) from the series:
\[ 1 + 0 = 1 \]

### Final Answer
Thus, the value of the series \(1 + i^2 + i^4 + i^6 + \ldots + i^{100}\) is:
\[ \boxed{1} \]