Evaluate the Summation sum from i=1 to 60 of -4i
The question is asking for the evaluation of a mathematical summation. Specifically, it requires the calculation of the sum of terms formed by multiplying each integer from 1 to 60 by -4. The task involves applying the concept of series and sequences, particularly arithmetic series, where each term in the sequence is a constant difference away from its neighboring terms. The problem tests knowledge of summation formulas and algebraic manipulation to obtain a final, simplified result representing the sum of all the terms in this specific series.
$\sum_{i = 1}^{60} - 4 i$
Answer
Solution:
Step 1:
Extract the constant $-4$ from the summation expression: $-4 \sum_{i=1}^{60} i$
Step 2:
Utilize the arithmetic series sum formula: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
Step 3:
Insert the upper limit of the summation into the formula and include the extracted constant: $-4 \left( \frac{60(60+1)}{2} \right)$
Step 4:
Proceed with the simplification process:
Step 4.1:
Begin by simplifying the numerator:
Step 4.1.1:
Calculate $60 + 1$: $-4 \frac{60 \cdot 61}{2}$
Step 4.1.2:
Multiply $60$ by $61$: $-4 \left( \frac{3660}{2} \right)$
Step 4.2:
Reduce the fraction by eliminating common factors:
Step 4.2.1:
Isolate the factor of $2$ from $-4$: $2(-2) \frac{3660}{2}$
Step 4.2.2:
Eliminate the common factor of $2$: $\cancel{2} \cdot -2 \frac{3660}{\cancel{2}}$
Step 4.2.3:
Simplify the expression: $-2 \cdot 3660$
Step 4.3:
Complete the multiplication: $-2 \cdot 3660 = -7320$
Knowledge Notes:
The problem involves evaluating the summation of a linear sequence, which is a common task in algebra and calculus. The relevant knowledge points include:
Summation Notation: The summation symbol $\sum$ represents the sum of a sequence of numbers. The variable $i$ is the index of summation, and the limits of summation are specified below and above the summation symbol.
Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence in which each term after the first is obtained by adding a constant difference to the previous term. The sum of the first $n$ terms of an arithmetic sequence is given by the formula $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$.
Factoring Constants from Summations: When a constant multiplier is present in each term of the summation, it can be factored out and placed in front of the summation symbol, simplifying the calculation.
Simplification: The process of simplification involves performing arithmetic operations to reduce expressions to their simplest form. This can include adding or multiplying numbers, canceling common factors, and combining like terms.
Algebraic Manipulation: The solution requires manipulating algebraic expressions, including factoring, expanding, and simplifying fractions.
Understanding these concepts is essential for solving summation problems and for a wide range of mathematical applications.
