Evaluate the Summation sum from i=4 to 7 of 4^i

Solution:

Step 1: Expand the Summation

Calculate each term in the summation individually for $i = 4, 5, 6, 7$.

$4^4 + 4^5 + 4^6 + 4^7$

Step 2: Perform the Calculations

Step 2.1: Calculate $4^4$

Compute $4$ raised to the fourth power.

$256 + 4^5 + 4^6 + 4^7$

Step 2.2: Calculate $4^5$

Compute $4$ raised to the fifth power.

$256 + 1024 + 4^6 + 4^7$

Step 2.3: Combine the Results

Sum the first two terms.

$1280 + 4^6 + 4^7$

Step 2.4: Calculate $4^6$

Compute $4$ raised to the sixth power.

$1280 + 4096 + 4^7$

Step 2.5: Combine the Results

Sum the first three terms.

$5376 + 4^7$

Step 2.6: Calculate $4^7$

Compute $4$ raised to the seventh power.

$5376 + 16384$

Step 2.7: Combine the Results

Sum all the terms to get the final result.

$21760$

Knowledge Notes:

The problem involves evaluating a finite summation, which is a common operation in mathematics where you sum the values of a sequence of numbers. The sequence in this case is generated by the function $4^i$ for the consecutive integer values of $i$ from $4$ to $7$.

To solve the problem, we use the following knowledge points:

1. Exponentiation: This is the operation of raising a number to a power, which is the number of times it is multiplied by itself. For example, $4^3 = 4 \times 4 \times 4 = 64$.

2. Summation: This is the addition of a sequence of any kind of numbers, which in this case are powers of $4$. The summation symbol $\sum$ is used to represent the sum of a sequence of terms.

3. Arithmetic Operations: Basic arithmetic operations like addition are used to combine the terms after they have been calculated.

The process of solving the problem involves expanding the summation into its individual terms, calculating each term by raising $4$ to the respective power, and then adding all the terms together to find the total sum.