Evaluate the Summation sum from k=1 to n of (k+1)^2
The question is asking for an analysis of a mathematical series where each term in the series takes the form of the square of (k+1), with k starting at value 1 and increasing by 1 with each term until k equals n. The task is to perform a summation of all the values generated by this formula for each value of k within the given range. Essentially, you're being asked to find the total when you add together the squares of each integer from 2 (since k starts at 1, k+1 is 2) up to and including (n+1), because when k equals n, you're squaring (n+1).
$\sum_{k = 1}^{n} \left(\left(\right. k + 1 \left.\right)\right)^{2}$
Answer
Solution:
Step 1:
Express $(k + 1)^2$ as a product of two identical binomials: $(k + 1)(k + 1)$.
Step 2:
Expand the binomial product $(k + 1)(k + 1)$ by employing the FOIL (First, Outer, Inner, Last) technique.
Step 2.1:
Distribute the terms: $k(k + 1) + 1(k + 1)$.
Step 2.2:
Continue distribution: $k \cdot k + k \cdot 1 + 1 \cdot k + 1 \cdot 1$.
Step 2.3:
Complete the distribution: $k^2 + k + k + 1$.
Step 3:
Combine like terms and simplify the expression.
Step 3.1:
Simplify each term individually.
Step 3.1.1:
Multiply $k$ by itself: $k^2 + k + k + 1$.
Step 3.1.2:
Multiply $k$ by $1$: $k^2 + k + k + 1$.
Step 3.1.3:
Multiply $1$ by $k$: $k^2 + k + k + 1$.
Step 3.1.4:
Multiply $1$ by $1$: $k^2 + k + k + 1$.
Step 3.2:
Combine the terms $k$ and $k$: $k^2 + 2k + 1$.
Step 4:
Represent the summation with the simplified expression: $\sum_{k=1}^{n} (k^2 + 2k + 1)$.
Knowledge Notes:
The problem involves evaluating the sum of the squares of consecutive integers, starting from 1 and ending at $n$, with each term incremented by 1 before squaring. The solution requires knowledge of algebraic manipulation, specifically:
Binomial Expansion: The process of expanding expressions that are raised to a power, such as $(k + 1)^2$. This involves multiplying the binomial by itself.
FOIL Method: A technique used to expand binomials, where you multiply the First terms, then the Outer terms, the Inner terms, and finally the Last terms of the binomial product.
Distributive Property: A property that allows one to distribute multiplication over addition or subtraction, such as in $a(b + c) = ab + ac$.
Combining Like Terms: The process of simplifying an algebraic expression by adding or subtracting terms that have the same variables raised to the same power.
Summation Notation: The use of the sigma symbol $\Sigma$ to denote the sum of a sequence of terms. The expression $\sum_{k=1}^{n} (k^2 + 2k + 1)$ represents the sum of the terms $(k^2 + 2k + 1)$ as $k$ varies from 1 to $n$.
Understanding these concepts is crucial for solving problems involving algebraic expressions and their summation.
