Let ` S = (2+6)/(4^(100)) + (2 + 2 xx 6)/(4^(99)) + (2 + 3xx 6)/(4^(98)) + …+ (2 + 99 xx6)/(4^(2)) + (2 + 100xx 6)/(4)` . Then 3S equals .

Answer

To solve the problem, we need to evaluate the sum \( S \) given by:

\[ S = \frac{2 + 6}{4^{100}} + \frac{2 + 2 \times 6}{4^{99}} + \frac{2 + 3 \times 6}{4^{98}} + \ldots + \frac{2 + 99 \times 6}{4^{2}} + \frac{2 + 100 \times 6}{4} \]

We can rewrite the terms in the series:

\[ S = \sum_{n=1}^{100} \frac{2 + n \cdot 6}{4^{101-n}} \]

This can be separated into two parts:

\[ S = \sum_{n=1}^{100} \frac{2}{4^{101-n}} + \sum_{n=1}^{100} \frac{n \cdot 6}{4^{101-n}} \]

Let’s denote the first sum as \( S_1 \) and the second sum as \( S_2 \).

### Step 1: Calculate \( S_1 \)

The first sum \( S_1 \) can be expressed as:

\[ S_1 = 2 \sum_{n=1}^{100} \frac{1}{4^{101-n}} = 2 \sum_{k=1}^{100} \frac{1}{4^k} \]

This is a geometric series with first term \( a = \frac{1}{4} \) and common ratio \( r = \frac{1}{4} \). The number of terms is 100.

The sum of a geometric series is given by:

\[ \text{Sum} = \frac{a(1 - r^n)}{1 - r} \]

Applying this formula:

\[ S_1 = 2 \cdot \frac{\frac{1}{4}(1 - (\frac{1}{4})^{100})}{1 - \frac{1}{4}} = 2 \cdot \frac{\frac{1}{4}(1 - \frac{1}{4^{100}})}{\frac{3}{4}} = \frac{2}{3}(1 - \frac{1}{4^{100}}) \]

### Step 2: Calculate \( S_2 \)

Now, we calculate \( S_2 \):

\[ S_2 = 6 \sum_{n=1}^{100} \frac{n}{4^{101-n}} = 6 \sum_{k=1}^{100} \frac{k}{4^k} \]

To find this sum, we can use the formula for the sum of \( n \cdot r^n \):

\[ \sum_{k=1}^{n} kx^k = \frac{x(1 - (n+1)x^n + nx^{n+1})}{(1-x)^2} \]

For our case, \( x = \frac{1}{4} \) and \( n = 100 \):

\[ S_2 = 6 \cdot \frac{\frac{1}{4}(1 - 101 \cdot \frac{1}{4^{100}} + 100 \cdot \frac{1}{4^{101}})}{(1 - \frac{1}{4})^2} \]

Calculating the denominator:

\[ (1 - \frac{1}{4})^2 = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

Thus,

\[ S_2 = 6 \cdot \frac{\frac{1}{4}(1 - 101 \cdot \frac{1}{4^{100}} + 100 \cdot \frac{1}{4^{101}})}{\frac{9}{16}} = 6 \cdot \frac{4}{9}(1 - 101 \cdot \frac{1}{4^{100}} + 100 \cdot \frac{1}{4^{101}}) \]

### Step 3: Combine \( S_1 \) and \( S_2 \)

Now, we can combine \( S_1 \) and \( S_2 \):

\[ S = S_1 + S_2 = \frac{2}{3}(1 - \frac{1}{4^{100}}) + 6 \cdot \frac{4}{9}(1 - 101 \cdot \frac{1}{4^{100}} + 100 \cdot \frac{1}{4^{101}}) \]

### Step 4: Calculate \( 3S \)

Finally, we need to find \( 3S \):

\[ 3S = 3 \left( \frac{2}{3}(1 - \frac{1}{4^{100}}) + 6 \cdot \frac{4}{9}(1 - 101 \cdot \frac{1}{4^{100}} + 100 \cdot \frac{1}{4^{101}}) \right) \]

This will give us the final answer.