Evaluate the Summation sum from i=1 to 4 of 4i+1
The problem provided is asking for the calculation of a specific arithmetic series. The series to be evaluated consists of terms that follow the rule 4i+1, with 'i' representing each integer starting from 1 and going to 4. The task is to find the sum of all terms generated by applying the rule for each value of 'i' within the given range. You are being asked to apply summation and arithmetic operations to determine the cumulative total when 'i' is equal to 1, 2, 3, and 4, respectively, and then add up those four results.
$\sum_{i = 1}^{4} 4 i + 1$
Answer
Solution:
Step 1:
Write out the terms of the summation for each value of $i$: $4 \cdot 1 + 1, 4 \cdot 2 + 1, 4 \cdot 3 + 1, 4 \cdot 4 + 1$.
Step 2:
Begin simplifying the expression.
Step 2.1:
Calculate $4 \times 1$: $4 + 1, 4 \cdot 2 + 1, 4 \cdot 3 + 1, 4 \cdot 4 + 1$.
Step 2.2:
Combine $4$ and $1$: $5, 4 \cdot 2 + 1, 4 \cdot 3 + 1, 4 \cdot 4 + 1$.
Step 2.3:
Calculate $4 \times 2$: $5, 8 + 1, 4 \cdot 3 + 1, 4 \cdot 4 + 1$.
Step 2.4:
Combine $8$ and $1$: $5, 9, 4 \cdot 3 + 1, 4 \cdot 4 + 1$.
Step 2.5:
Combine $5$ and $9$: $14, 4 \cdot 3 + 1, 4 \cdot 4 + 1$.
Step 2.6:
Calculate $4 \times 3$: $14, 12 + 1, 4 \cdot 4 + 1$.
Step 2.7:
Combine $12$ and $1$: $14, 13, 4 \cdot 4 + 1$.
Step 2.8:
Combine $14$ and $13$: $27, 4 \cdot 4 + 1$.
Step 2.9:
Calculate $4 \times 4$: $27, 16 + 1$.
Step 2.10:
Combine $16$ and $1$: $27, 17$.
Step 2.11:
Combine $27$ and $17$: $44$.
The final result is $44$.
Knowledge Notes:
To evaluate the summation $\sum_{i=1}^{4} (4i + 1)$, we follow a systematic approach:
Understanding the Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of terms. The variable $i$ is the index of summation, starting at the lower limit (here, 1) and increasing by 1 until it reaches the upper limit (here, 4).
Expanding the Summation: We substitute each value of $i$ from the lower limit to the upper limit into the expression $4i + 1$ to generate the terms of the series.
Simplifying Each Term: Each term of the form $4i + 1$ is simplified by performing the multiplication and then the addition.
Combining Like Terms: After simplifying, we combine the terms to find the total sum.
Arithmetic Operations: Basic arithmetic operations like multiplication and addition are used throughout the process.
Final Answer: After all terms have been combined, we arrive at the final answer.
In this problem, we are dealing with a finite arithmetic series, where each term is generated by a linear expression of the index $i$. The process of evaluating the summation involves expanding, simplifying, and then summing up all the terms.
