Evaluate the Summation sum from n=11 to 13 of 8n^2
The question asks for the calculation of a finite summation. Specifically, it requests the evaluation of a sum where the summation index, denoted by n, starts at 11 and ends at 13, incrementing by 1 each time. The function to be summed is 8n^2 for each value of n within the given range. You are expected to substitute the values 11, 12, and 13 into the function 8n^2, compute the square for each n, multiply the result by 8, and then add all the computed values together to find the final sum.
$\sum_{n = 11}^{13} 8 n^{2}$
Answer
Solution:
Step 1:
Write out the sum for each term where $n$ ranges from $11$ to $13$.
$8 \cdot 11^2 + 8 \cdot 12^2 + 8 \cdot 13^2$
Step 2:
Perform the calculations step by step.
Step 2.1:
Square the number $11$.
$8 \cdot 121 + 8 \cdot 12^2 + 8 \cdot 13^2$
Step 2.2:
Multiply the square of $11$ by $8$.
$968 + 8 \cdot 12^2 + 8 \cdot 13^2$
Step 2.3:
Square the number $12$.
$968 + 8 \cdot 144 + 8 \cdot 13^2$
Step 2.4:
Multiply the square of $12$ by $8$.
$968 + 1152 + 8 \cdot 13^2$
Step 2.5:
Combine the first two terms.
$2120 + 8 \cdot 13^2$
Step 2.6:
Square the number $13$.
$2120 + 8 \cdot 169$
Step 2.7:
Multiply the square of $13$ by $8$.
$2120 + 1352$
Step 2.8:
Add the last two terms to get the final result.
$3472$
Knowledge Notes:
The problem involves evaluating a finite summation where the summand is a quadratic function of the variable $n$. The process of solving such a problem typically involves the following knowledge points:
Understanding of Summation Notation: The summation notation $\sum$ is a shorthand used to represent the sum of a sequence of terms. The variable $n$ is the index of summation, and the limits of summation indicate the range over which $n$ varies.
Arithmetic Operations: Basic arithmetic operations such as exponentiation (raising to a power), multiplication, and addition are used to evaluate each term in the series.
Exponentiation: Squaring a number means multiplying the number by itself ($n^2 = n \cdot n$).
Multiplication by a Constant: Each term in the series is multiplied by the constant $8$. This is a distributive property where $8 \cdot n^2$ is evaluated for each value of $n$.
Sequential Calculation: The problem is solved by performing calculations in a sequence, starting with squaring each number, then multiplying by the constant, and finally adding up all the terms.
Final Summation: After calculating each term, the final step is to sum all the terms to get the result of the summation.
By following these steps, the summation of the series is found to be $3472$.
