Evaluate the Summation sum from i=1 to 4 of 4i+1

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:

1. 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).

2. 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.

3. Simplifying Each Term: Each term of the form $4i+1$ is simplified by performing the multiplication and then the addition.

4. Combining Like Terms: After simplifying, we combine the terms to find the total sum.

5. Arithmetic Operations: Basic arithmetic operations like multiplication and addition are used throughout the process.

6. 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.