Evaluate the Summation sum from i=1 to 4 of i(i-2)

Solution:

Step:1

Decompose the given summation expression.

Step:1.1

Utilize the distributive law to expand: $i \cdot i + i \cdot (-2)$

Step:1.2

Perform the multiplication of $i$ with itself: $i^2 + i \cdot (-2)$

Step:1.3

Rearrange the terms to place the negative in front: $i^2 - 2i$

Step:1.4

Express the summation with the new terms: $\sum_{i = 1}^{4} (i^2 - 2i)$

Step:2

Calculate the series by inserting each value of $i$: $1^2 - 2 \cdot 1 + 2^2 - 2 \cdot 2 + 3^2 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3

Simplify the expression step by step.

Step:3.1

Square the number $1$: $1 - 2 \cdot 1 + 2^2 - 2 \cdot 2 + 3^2 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3.2

Multiply $-2$ by $1$: $1 - 2 + 2^2 - 2 \cdot 2 + 3^2 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3.3

Combine $1$ and $-2$: $-1 + 2^2 - 2 \cdot 2 + 3^2 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3.4

Square the number $2$: $-1 + 4 - 2 \cdot 2 + 3^2 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3.5

Multiply $-2$ by $2$: $-1 + 4 - 4 + 3^2 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3.6

Combine $4$ and $-4$: $-1 + 0 + 3^2 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3.7

Add $-1$ to $0$: $-1 + 3^2 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3.8

Square the number $3$: $-1 + 9 - 2 \cdot 3 + 4^2 - 2 \cdot 4$

Step:3.9

Multiply $-2$ by $3$: $-1 + 9 - 6 + 4^2 - 2 \cdot 4$

Step:3.10

Combine $9$ and $-6$: $-1 + 3 + 4^2 - 2 \cdot 4$

Step:3.11

Add $-1$ to $3$: $2 + 4^2 - 2 \cdot 4$

Step:3.12

Square the number $4$: $2 + 16 - 2 \cdot 4$

Step:3.13

Multiply $-2$ by $4$: $2 + 16 - 8$

Step:3.14

Combine $16$ and $-8$: $2 + 8$

Step:3.15

Add $2$ to $8$: $10$

Knowledge Notes:

To solve the given problem, we use the following knowledge points:

1. Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of numbers. The expression $\sum_{i = a}^{b} f(i)$ means the sum of $f(i)$ for all integer values of $i$ from $a$ to $b$ inclusive.

2. Distributive Property: This property allows us to multiply a sum by a number by multiplying each addend of the sum by the number. For example, $a(b + c) = ab + ac$.

3. Arithmetic Operations: Basic arithmetic operations include addition, subtraction, multiplication, and exponentiation. These operations are used to simplify expressions.

4. Simplifying Expressions: This involves combining like terms and performing arithmetic operations to reduce an expression to its simplest form.

5. Exponentiation: Raising a number to a power, denoted as $a^n$, means multiplying the number $a$ by itself $n$ times.

By understanding and applying these concepts, we can evaluate the given summation expression step by step to reach the final result.