If i = √-1 What is the value of 1+ i + i2 + i3+ ...... + i100

The given series is a geometric series with a common ratio of i.

The sum of a geometric series is given by the formula:

S = a * (1 - r^n) / (1 - r)

where:

• S is the sum of the series
• a is the first term of the series
• r is the common ratio of the series
• n is the number of terms in the series

In this case, a = 1, r = i, and n = 101.

However, we know that i^2 = -1, i^3 = -i, and i^4 = 1.

So, the powers of i repeat every 4 terms: i, -1, -i, 1, i, -1, -i, 1, ...

Therefore, we can simplify the series to:

1 + i - 1 - i + 1 + i - 1 - i + ... + 1

Since there are 101 terms in the series, and the pattern repeats every 4 terms, there are 25 full repetitions of the pattern and one extra term.

Each full repetition of the pattern sums to 0 (since 1 + i - 1 - i = 0), so the sum of the first 100 terms is 0.

The 101st term is 1 (since it's the first term of the 26th repetition of the pattern), so the sum of the series is 0 + 1 = 1.

Therefore, the value of 1+ i + i^2 + i^3+ ...... + i^100 is 1.