1/3-2/3^2-3/3^34/3^4 +...+ 99/3^99-100/3^100 < 1/4

To solve this inequality, we can first simplify the expression on the left side:

1/3 - 2/3^2 - 3/3^3 + ... + 99/3^99 - 100/3^100

Notice that each term can be written as n/3^n, where n ranges from 1 to 100. We can rewrite this expression as:

∑(n/3^n) from n=1 to 100

Now, let's find a common denominator for all the terms:

∑(n/3^n) = (∑n * 3^100 + ∑(n * 3^99) + ∑(n * 3^98) + ... + ∑(n * 3^1)) / 3^100

Using the formula for the sum of an arithmetic series, we can simplify each term:

∑n = (100 * (100 + 1)) / 2 = 5050

∑(n * 3^k) = (100 * (100 + 1) * 3^k) / 2 = 5050 * 3^k

Now, let's substitute these values back into the expression:

(5050 * 3^100 + 5050 * 3^99 + 5050 * 3^98 + ... + 5050 * 3^1) / 3^100

Factoring out 5050:

5050 * (3^100 + 3^99 + 3^98 + ... + 3^1) / 3^100

Simplifying the numerator:

3^100 + 3^99 + 3^98 + ... + 3^1 = (3^101 - 3) / (3 - 1) = (3^101 - 3) / 2

Substituting back into the expression:

5050 * (3^101 - 3) / (2 * 3^100)

Simplifying further:

5050 * (3^101 - 3) / (2 * 3^100) = 5050 * (3^101 / 2 * 3^100 - 3 / 2 * 3^100) = 5050 * (3/2 - 3/2 * 1/3^100)

Now, let's evaluate this expression:

5050 * (3/2 - 3/2 * 1/3^100) = 5050 * (3/2 - 1/2 * 1/3^99) = 5050 * (3/2 - 1/2 * 1/3^99) = 5050 * (3/2 - 1/2 * 1/3^99)

Since 1/3^99 is a very small number, we can approximate it as 0:

5050 * (3/2 - 1/2 * 0) = 5050 * (3/2) = 7575

Therefore, the expression is greater than 7575, which is greater than 1/4.