Evaluate the Summation sum from k=1 to 15 of 5+3(k-1)
The question is asking for the evaluation of a finite arithmetic series. Specifically, you are required to calculate the sum of the terms in the sequence generated by the formula 5 + 3(k - 1), as k increases from 1 to 15. Each term in the series is determined by substituting the value of k into the formula, and the summation sign indicates that you should find the total sum of these individual terms within the specified range of k.
$\sum_{k = 1}^{15} 5 + 3 \left(\right. k - 1 \left.\right)$
Answer
Solution:
Step 1: Simplify the given summation expression.
Step 1.1: Break down the summation term.
Step 1.1.1: Use the distributive property to expand: $5 + 3(k - 1)$.
Step 1.1.2: Perform the multiplication: $5 + 3k - 3$.
Step 1.2: Combine like terms: $3k + 2$.
Step 1.3: Represent the simplified summation: $\sum_{k=1}^{15}(3k + 2)$.
Step 2: Decompose the summation into two separate summations.
- $\sum_{k=1}^{15}(3k + 2) = 3\sum_{k=1}^{15}k + \sum_{k=1}^{15}2$.
Step 3: Calculate the summation of the linear term $3\sum_{k=1}^{15}k$.
Step 3.1: Use the formula for the sum of the first $n$ natural numbers: $\sum_{k=1}^{n}k = \frac{n(n + 1)}{2}$.
Step 3.2: Plug in the upper limit of the summation and include the coefficient: $3\left(\frac{15(15 + 1)}{2}\right)$.
Step 3.3: Simplify the expression.
Step 3.3.1: Add the numbers inside the parentheses: $3\frac{15 \cdot 16}{2}$.
Step 3.3.2: Multiply the numbers: $3\left(\frac{240}{2}\right)$.
Step 3.3.3: Divide by $2$: $3 \cdot 120$.
Step 3.3.4: Multiply by $3$: $360$.
Step 4: Calculate the summation of the constant term $\sum_{k=1}^{15}2$.
Step 4.1: Apply the formula for the sum of a constant: $\sum_{k=1}^{n}c = cn$.
Step 4.2: Substitute the values: $(2)(15)$.
Step 4.3: Perform the multiplication: $30$.
Step 5: Combine the results of the two summations.
- $360 + 30$.
Step 6: Compute the final sum.
- $390$.
Knowledge Notes:
To solve the given problem, we need to understand several key concepts in algebra and arithmetic:
Distributive Property: This property allows us to multiply a single term by each term within a parenthesis. For example, $a(b + c) = ab + ac$.
Summation (Sigma Notation): Summation is a way to add up a series of numbers. The sigma notation $\sum$ is used to represent the sum of a sequence of terms.
Sum of the First n Natural Numbers: The formula for the sum of the first $n$ natural numbers is given by $\sum_{k=1}^{n}k = \frac{n(n + 1)}{2}$.
Sum of a Constant: The sum of a constant $c$ over $n$ terms is simply $cn$.
Combining Like Terms: In algebra, combining like terms is a basic process to simplify expressions. Terms that have the same variable part can be combined by adding or subtracting their coefficients.
By applying these concepts, we can simplify the original summation expression, break it down into more manageable parts, calculate each part using the appropriate formulas, and then combine the results to find the final sum.
