Evaluate the Summation sum from k=1 to n of (k+1)^2

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:

1. 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.

2. 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.

3. Distributive Property: A property that allows one to distribute multiplication over addition or subtraction, such as in $a(b+c)=ab+ac$.

4. 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.

5. Summation Notation: The use of the sigma symbol $\mathrm{\Sigma }$ 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.