Evaluate the Summation sum from i=1 to 5 of 3^i-4
The question is asking for a mathematical evaluation of a summation, which means you are expected to calculate the total sum of a series that runs from i equal to 1 up to i equal to 5. In this series, the term to sum is provided by the expression 3^i - 4 for each value of i within the given range. Here, 3^i signifies three raised to the power of i, and from that result, you subtract the number 4. The task involves performing this calculation for each individual i (from 1 to 5) and then combining all of these results to obtain the final sum.
$\sum_{i = 1}^{5} 3^{i} - 4$
Answer
Solution:
Step 1:
Write out the terms of the series by substituting $i$ with each integer from 1 to 5.
$3^{1} - 4, 3^{2} - 4, 3^{3} - 4, 3^{4} - 4, 3^{5} - 4$
Step 2:
Calculate the sum of the terms.
$3^{1} - 4 + 3^{2} - 4 + 3^{3} - 4 + 3^{4} - 4 + 3^{5} - 4 = 343$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a series of terms added together. The summation is defined by a formula that depends on an index variable, in this case, $i$. The index variable ranges from a lower bound (1) to an upper bound (5), and for each value of $i$, a term is generated and included in the summation.
The formula for the terms of the series in this problem is $3^{i} - 4$. This means that for each value of $i$, we calculate $3$ raised to the power of $i$ and then subtract $4$.
To solve the problem, we follow these steps:
Expansion: We expand the series by calculating the value of each term using the given formula for all values of $i$ within the bounds of the summation. In this case, we calculate the terms for $i = 1, 2, 3, 4, 5$.
Simplification: After expanding the series, we add up all the terms to get the final sum. This involves performing the exponentiation and subtraction for each term and then summing all the results.
In this particular problem, the simplification step directly gives the numerical result of the summation, which is $343$. This is because the problem does not require us to express the sum in a simplified algebraic form but rather to calculate the actual numerical sum.
Relevant mathematical concepts include:
Summation notation ($\Sigma$), which is used to represent the sum of a sequence of terms.
Exponentiation, which is the operation of raising one number (the base) to the power of another number (the exponent).
Arithmetic operations, such as addition and subtraction, which are used to combine the terms of the series.
Understanding these concepts is essential for solving problems involving summations and series.
