Find the value of 1+i^{2}+i^{4}+i^{6}+i^{8}+...+i^{20}

Are the term in AP find the Tn and Sn

Let's first analyze the given expression:

\[
1 + i^2 + i^4 + i^6 + i^8 + \cdots + i^{20}
\]

### Step 1: Properties of powers of \( i \)

The powers of the imaginary unit \( i \) repeat in a cycle of 4:

- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)

Thus, powers of \( i \) repeat every 4 terms: \( i^5 = i \), \( i^6 = -1 \), and so on.

### Step 2: Identifying the terms

Now, let’s calculate each power of \( i \) for the given sequence:

- \( i^0 = 1 \)
- \( i^2 = -1 \)
- \( i^4 = 1 \)
- \( i^6 = -1 \)
- \( i^8 = 1 \)
- \( i^{10} = -1 \)
- \( i^{12} = 1 \)
- \( i^{14} = -1 \)
- \( i^{16} = 1 \)
- \( i^{18} = -1 \)
- \( i^{20} = 1 \)

### Step 3: Grouping the terms

The sum is:

\[
1 + (-1) + 1 + (-1) + 1 + (-1) + 1 + (-1) + 1 + (-1) + 1 = 1
\]

So the value of the sum is \( \boxed{1} \).

### Step 4: Are the terms in Arithmetic Progression (AP)?

No, the terms are not in an arithmetic progression (AP). In an AP, the difference between consecutive terms is constant, but here the terms alternate between 1 and -1.

### Step 5: General Term (Tn) and Sum of n terms (Sn)

Since the terms are not in AP, we cannot directly use the AP formula to find \( T_n \) and \( S_n \). However, we can define a pattern:

- For \( n \) even (i.e., \( i^{2n} \)), the term is \( 1 \).
- For \( n \) odd (i.e., \( i^{2n+1} \)), the term is \( -1 \).

Thus, for the sum \( S_n \), we see that alternating pairs of \( 1 + (-1) = 0 \), leaving us with just the first term \( i^0 = 1 \).

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Would you like more details or have any questions? Here are five related questions:

1. What is the cycle of powers of \( i \) beyond \( i^4 \)?
2. How do we classify imaginary and real numbers in this context?
3. What is an arithmetic progression (AP), and how do we determine one?
4. Can we sum alternating sequences in other ways?
5. What other types of sequences can imaginary numbers form?

**Tip:** Powers of imaginary numbers repeat in cycles, which is helpful in summing complex expressions.

Find the value of 1 + i^2 + i^4 + i^6 + i^8 + ... + i^20. Are the terms in AP? Find the Tn and Sn.

Explore how to sum the series 1 + i^2 + i^4 + ... + i^20 using the cyclic nature of powers of i. This detailed solution breaks down the steps and investigates if the terms form an arithmetic progression. Suitable for Grades 9-12.

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