Evaluate the Summation sum from i=1 to n of (5+2i)^2
The question asks for an evaluation of the finite sum of the squares of terms in the form of (5+2i) for every integer value of i from 1 to n inclusive. Specifically, the expression to be summed is (5+2i)^2, which should be expanded and simplified. The summation operator is applied to this quadratic expression, and it indicates that we need to calculate the sum by plugging in each integer value of i within the range and then adding all the resulting terms together. The final answer would be a function of n, as n determines the number of terms in the sum.
$\sum_{i = 1}^{n} \left(\left(\right. 5 + 2 i \left.\right)\right)^{2}$
Answer
Solution:
Step 1
Transform $(5 + 2i)^2$ into $(5 + 2i)(5 + 2i)$.
Step 2
Use the FOIL (First, Outer, Inner, Last) method to expand $(5 + 2i)(5 + 2i)$.
Step 2.1
Distribute the first term: $5(5 + 2i) + 2i(5 + 2i)$.
Step 2.2
Distribute the $5$: $5 \cdot 5 + 5(2i) + 2i(5 + 2i)$.
Step 2.3
Continue distribution: $5 \cdot 5 + 5(2i) + 2i \cdot 5 + 2i(2i)$.
Step 3
Combine like terms and simplify.
Step 3.1
Perform the multiplication for each term.
Step 3.1.1
Calculate $5 \cdot 5$: $25 + 5(2i) + 2i \cdot 5 + 2i(2i)$.
Step 3.1.2
Multiply $5$ and $2$: $25 + 10i + 2i \cdot 5 + 2i(2i)$.
Step 3.1.3
Multiply $2i$ and $5$: $25 + 10i + 10i + 2i(2i)$.
Step 3.1.4
Apply the commutative property: $25 + 10i + 10i + 4i^2$.
Step 3.1.5
Simplify the power of $i$.
Step 3.1.5.1
Rearrange the terms: $25 + 10i + 10i + 4(i \cdot i)$.
Step 3.1.5.2
Calculate $i \cdot i$: $25 + 10i + 10i + 4i^2$.
Step 3.1.6
Multiply $4$ and $i^2$: $25 + 10i + 10i + 4i^2$.
Step 3.2
Combine the $10i$ terms: $25 + 20i + 4i^2$.
Step 4
Express the summation: $\sum_{i=1}^{n} (25 + 20i + 4i^2)$.
Knowledge Notes:
To solve the given problem, we need to understand several mathematical concepts and methods:
Binomial Expansion: The process of expanding expressions that are raised to a power, in this case, $(5 + 2i)^2$.
FOIL Method: A technique for expanding binomials, where you multiply the First, Outer, Inner, and Last terms of the binomial.
Distributive Property: A property that allows us to multiply a sum by multiplying each addend separately and then sum the products.
Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product, i.e., $ab = ba$.
Combining Like Terms: The process of simplifying expressions by adding or subtracting terms that have the same variable raised to the same power.
Summation Notation: The use of the sigma symbol $\Sigma$ to represent the sum of a sequence of terms. In this case, $\sum_{i=1}^{n}$ denotes the sum of terms for $i$ ranging from $1$ to $n$.
Arithmetic Operations: Basic operations such as addition, subtraction, multiplication, and exponentiation are used throughout the problem-solving process.
By applying these concepts and methods, we can expand the given expression, simplify it, and then represent the sum of the sequence in summation notation.
