Evaluate the Summation sum from x=2 to 5 of 2^x+3x-6
The question provided asks for the calculation of a summation where the index variable x starts at 2 and ends at 5. For each value of x in this range, you are to compute the expression 2^x + 3x - 6 and then sum all the resulting values together. Essentially, you need to evaluate this expression four times, once for each integer value of x from 2 to 5, and then add up all those computed values to find the final sum.
$\sum_{x = 2}^{5} 2^{x} + 3 x - 6$
Answer
Solution:
Step:1
List out the terms of the summation for each integer value of $x$ from $2$ to $5$.
$$2^2 + 3 \cdot 2 - 6 + 2^3 + 3 \cdot 3 - 6 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2
Proceed to simplify the expression.
Step:2.1
Calculate $2$ raised to the power of $2$.
$$4 + 3 \cdot 2 - 6 + 2^3 + 3 \cdot 3 - 6 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.2
Compute the product of $3$ and $2$.
$$4 + 6 - 6 + 2^3 + 3 \cdot 3 - 6 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.3
Combine $4$ and $6$.
$$10 - 6 + 2^3 + 3 \cdot 3 - 6 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.4
Subtract $6$ from $10$.
$$4 + 2^3 + 3 \cdot 3 - 6 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.5
Calculate $2$ raised to the power of $3$.
$$4 + 8 + 3 \cdot 3 - 6 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.6
Compute the product of $3$ and $3$.
$$4 + 8 + 9 - 6 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.7
Combine $8$ and $9$.
$$4 + 17 - 6 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.8
Subtract $6$ from $17$.
$$4 + 11 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.9
Combine $4$ and $11$.
$$15 + 2^4 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.10
Calculate $2$ raised to the power of $4$.
$$15 + 16 + 3 \cdot 4 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.11
Compute the product of $3$ and $4$.
$$15 + 16 + 12 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.12
Combine $16$ and $12$.
$$15 + 28 - 6 + 2^5 + 3 \cdot 5 - 6$$
Step:2.13
Subtract $6$ from $28$.
$$15 + 22 + 2^5 + 3 \cdot 5 - 6$$
Step:2.14
Combine $15$ and $22$.
$$37 + 2^5 + 3 \cdot 5 - 6$$
Step:2.15
Calculate $2$ raised to the power of $5$.
$$37 + 32 + 3 \cdot 5 - 6$$
Step:2.16
Compute the product of $3$ and $5$.
$$37 + 32 + 15 - 6$$
Step:2.17
Combine $32$ and $15$.
$$37 + 47 - 6$$
Step:2.18
Subtract $6$ from $47$.
$$37 + 41$$
Step:2.19
Combine $37$ and $41$.
$$78$$
The final result is $78$.
Knowledge Notes:
To solve the given summation problem, we must understand the following concepts:
Summation Notation: The summation notation $\sum$ represents the sum of a sequence of numbers. The expression under the summation symbol indicates the pattern of the sequence, while the limits of summation tell us where to start and end the sum.
Exponents: Raising a number to a power, denoted as $a^n$, means multiplying the number $a$ by itself $n$ times.
Arithmetic Operations: Basic arithmetic operations such as addition, subtraction, multiplication, and division are used to simplify expressions.
Order of Operations: When simplifying expressions, it is important to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Substitution: In the context of summation, substitution involves replacing the variable in the expression with each of the values in the range of summation and then performing the necessary calculations.
By applying these concepts, we can evaluate the summation by expanding the series, simplifying each term, and then adding the results to find the final sum.
