Evaluate the Summation sum from k=1 to 30 of k^2+2
The question is asking to perform a mathematical calculation, specifically to find the sum of a series where each term in the series is given by the expression \( k^2 + 2 \). The series starts with \( k = 1 \), and each term is to be calculated and added together until \( k \) reaches 30. Therefore, you are required to evaluate the sum of thirty terms, with the value of \( k \) increasing by 1 for each term, starting at \( k = 1 \) and ending at \( k = 30 \).
$\sum_{k = 1}^{30} k^{2} + 2$
Answer
Solution:
Step 1: Decompose the given summation into two separate summations.
The original summation can be broken down as follows:
$$\sum_{k = 1}^{30} (k^{2} + 2) = \sum_{k = 1}^{30} k^{2} + \sum_{k = 1}^{30} 2$$
Step 2: Calculate the summation of the squares of the first 30 natural numbers.
Step 2.1: Utilize the known formula for the sum of squares.
The formula to sum the squares of the first $n$ natural numbers is:
$$\sum_{k = 1}^{n} k^{2} = \frac{n(n + 1)(2n + 1)}{6}$$
Step 2.2: Insert the upper limit of the summation into the formula.
Plugging in the value of $n = 30$ into the formula gives us:
$$\frac{30(30 + 1)(2 \cdot 30 + 1)}{6}$$
Step 2.3: Simplify the expression.
Step 2.3.1: Reduce the fraction by eliminating common factors.
Extract the factor of 6 from the numerator:
$$\frac{6(5(30 + 1)(2 \cdot 30 + 1))}{6}$$
Step 2.3.1.2: Remove the common factors.
Divide the numerator and the denominator by 6:
$$\frac{6(5(30 + 1)(2 \cdot 30 + 1))}{6 \cdot 1}$$
Step 2.3.1.2.3: Rewrite the simplified expression.
After canceling out the 6, we are left with:
$$\frac{5(30 + 1)(2 \cdot 30 + 1)}{1}$$
Step 2.3.2: Perform the arithmetic operations.
Add, multiply, and simplify the terms:
$$5 \cdot 31(2 \cdot 30 + 1)$$ $$155(60 + 1)$$ $$155 \cdot 61$$ $$9455$$
Step 3: Calculate the summation of the constant term.
Step 3.1: Apply the formula for the sum of a constant over a range.
The formula for summing a constant $c$ over $n$ terms is:
$$\sum_{k = 1}^{n} c = c \cdot n$$
Step 3.2: Substitute the values into the formula.
For our constant $2$ over $30$ terms, the calculation is:
$$(2)(30)$$
Step 3.3: Multiply the constant by the number of terms.
$$2 \cdot 30$$ $$60$$
Step 4: Combine the results from both summations.
Add the sum of squares to the sum of the constant terms:
$$9455 + 60$$
Step 5: Compute the final sum.
$$9515$$
Knowledge Notes:
To solve the given problem, we applied several mathematical concepts and formulas:
Summation Decomposition: The ability to break down a summation into simpler parts that are easier to calculate.
Sum of Squares Formula: A specific formula used to calculate the sum of the squares of the first $n$ natural numbers, which is $\sum_{k = 1}^{n} k^{2} = \frac{n(n + 1)(2n + 1)}{6}$.
Summation of a Constant: The sum of a constant $c$ repeated $n$ times is simply $c$ multiplied by $n$, or $\sum_{k = 1}^{n} c = c \cdot n$.
Arithmetic Operations: Basic operations such as addition, multiplication, and simplification are used to calculate the final result.
Factorization and Reduction: The process of reducing fractions by canceling out common factors in the numerator and denominator to simplify calculations.
By applying these concepts, we were able to evaluate the given summation step by step and arrive at the final result of 9515.
