Evaluate the Summation sum from i=1 to 5 of 4i-9
The question is asking for the calculation of a finite summation. You are required to sum up the values of the expression 4i - 9 for each integer value of i starting from 1 and ending at 5. To solve this problem, you would plug in each value of i into the expression 4i - 9, calculate the result, and then add all of these results together to get the final answer. The summation represents a series of arithmetic operations based on a pattern described by the given formula.
$\sum_{i = 1}^{5} 4 i - 9$
Answer
Solution:
Step 1: Expansion of the Summation
Write out the terms of the summation for each integer value of $i$ from 1 to 5. The series becomes:
$4 \cdot 1 - 9 + 4 \cdot 2 - 9 + 4 \cdot 3 - 9 + 4 \cdot 4 - 9 + 4 \cdot 5 - 9$
Step 2: Calculation of the Sum
Compute the sum of the series by adding all the terms together to get the result:
$-15$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a common operation in mathematics where you add up a sequence of numbers generated by a formula involving an index variable. The index variable here is $i$, which takes on integer values from a starting point (1 in this case) to an endpoint (5 in this case). The formula given is $4i - 9$, which means for each value of $i$, you multiply it by 4 and subtract 9.
To solve this, you follow these steps:
Expansion: You substitute each integer value of $i$ into the formula to generate each term of the sequence. This step is crucial because it lays out all the terms that you need to add together.
Simplification: After expanding the series, you simplify by performing the arithmetic operations: multiplication and subtraction for each term, and then addition to combine all terms into a single sum.
The summation can also be approached by recognizing patterns or using formulas for arithmetic series, but since this is a simple case with only five terms, direct calculation is straightforward.
