Evaluate the Summation sum from i=1 to 5 of 4i+3
The question asks for the evaluation of a finite mathematical summation. Specifically, it requires the calculation of the sum of a sequence of terms generated by the expression 4i+3, where i represents each whole number from 1 to 5. The summation operator (denoted by the Greek letter Sigma, ∑) indicates that you are to add together the values of 4i+3 for each value of i within the specified range. To solve the question, one would calculate the expression for each individual value of i starting at i=1 and ending at i=5, and then sum all of these values together to get the final result.
$\sum_{i = 1}^{5} 4 i + 3$
Answer
Solution:
Step 1:
Write out the terms of the sequence by substituting $i$ with each integer from 1 to 5.
$4 \cdot 1 + 3, 4 \cdot 2 + 3, 4 \cdot 3 + 3, 4 \cdot 4 + 3, 4 \cdot 5 + 3$
Step 2:
Calculate the sum of the terms.
$4 \cdot 1 + 3 + 4 \cdot 2 + 3 + 4 \cdot 3 + 3 + 4 \cdot 4 + 3 + 4 \cdot 5 + 3 = 75$
Knowledge Notes:
To solve the given problem, we need to understand the concept of summation notation (also known as sigma notation). The summation notation is a way to represent the sum of a sequence of terms. The general form is $\sum_{i=m}^{n} a_i$, where $i$ is the index of summation, $m$ is the lower bound, $n$ is the upper bound, and $a_i$ is the general term of the sequence.
In this problem, we are asked to evaluate the sum of the sequence for $i$ ranging from 1 to 5, with the general term given by $4i + 3$. This means we need to substitute the index $i$ with each integer from 1 to 5, calculate each term, and then sum them all together.
The process involves two main steps:
Expansion: We expand the summation by writing out each term of the sequence explicitly. This step requires substituting the index $i$ with each value in the given range and writing out the resulting expression.
Simplification: After expanding the sequence, we simplify by performing the arithmetic operations to find the sum of all terms.
In this case, the arithmetic sequence is linear, and each term can be found by multiplying $i$ by 4 and then adding 3. After expanding and simplifying, we find that the sum of the sequence from $i=1$ to $i=5$ is 75.
