Evaluate the Summation sum from x=1 to 420 of 10x-54
The problem requires you to perform a summation calculation. Specifically, you are asked to calculate the total sum of an arithmetic sequence where each term is given by the formula 10x - 54. The sequence starts when x equals 1 and you continue adding subsequent terms until x equals 420. You are essentially being asked to add up all individual values generated by substituting x into 10x - 54, starting at x=1 and ending at x=420.
$\sum_{x = 1}^{420} 10 x - 54$
Answer
Solution:
Step:1
Decompose the given summation into two separate summations by distributing the summation over the subtraction.$\sum_{x = 1}^{420} (10x - 54) = 10 \sum_{x = 1}^{420} x - \sum_{x = 1}^{420} 54$
Step:2
Calculate the summation $10 \sum_{x = 1}^{420} x$.
Step:2.1
Apply the arithmetic series formula:$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$
Step:2.2
Insert the upper limit of the summation into the formula and include the coefficient.$10 \left( \frac{420(420 + 1)}{2} \right)$
Step:2.3
Proceed with the simplification.
Step:2.3.1
Begin simplifying the expression.
Step:2.3.1.1
Combine $420$ and $1$.$10 \frac{420 \cdot 421}{2}$
Step:2.3.1.2
Perform the multiplication of $420$ and $421$.$10 \left( \frac{176820}{2} \right)$
Step:2.3.2
Eliminate the common factor of $2$.
Step:2.3.2.1
Extract the factor of $2$ from the coefficient.$2(5) \frac{176820}{2}$
Step:2.3.2.2
Simplify by canceling out the common factor.$\cancel{2} \cdot 5 \frac{176820}{\cancel{2}}$
Step:2.3.2.3
Rewrite the simplified expression.$5 \cdot 176820$
Step:2.3.3
Complete the multiplication of $5$ and $176820$.$884100$
Step:3
Compute the summation $\sum_{x = 1}^{420} -54$.
Step:3.1
Use the formula for the summation of a constant term:$\sum_{k = 1}^{n} c = cn$
Step:3.2
Place the values into the formula.$(-54)(420)$
Step:3.3
Carry out the multiplication of $-54$ and $420$.$-22680$
Step:4
Combine the results from the two summations.$884100 - 22680$
Step:5
Subtract $22680$ from $884100$ to find the final result.$861420$
Solution:"861420"
Knowledge Notes:
Summation of an Arithmetic Series: The sum of the first $n$ natural numbers is given by the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$. This is an arithmetic series where each term increases by a constant difference.
Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $cn$, because each term in the summation is the same.
Distributive Property of Summation: The summation operator can be distributed over addition or subtraction, allowing us to break down complex summations into simpler parts.
Multiplication by a Summation: When a constant is multiplied by a summation, it can be factored out and multiplied by the result of the summation.
Simplification: Simplifying expressions often involves performing arithmetic operations like addition, multiplication, and canceling common factors.
By understanding these principles, one can tackle a wide range of summation problems in mathematics.
