Evaluate the Summation sum from i=1 to 5 of 4(-3)^i
The question asks for the evaluation of a finite mathematical summation where the sum is over the integers i from 1 to 5. In this summation, the summand is a function of i, specifically 4 times negative 3 raised to the power of i. To solve this problem, you would need to calculate each term by substituting i with the integers from 1 to 5 into the summand and then sum all the resulting terms together to find the total sum.
$\sum_{i = 1}^{5} 4 \left(\left(\right. - 3 \left.\right)\right)^{i}$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer value of $i$ from 1 to 5.
$4(-3)^1 + 4(-3)^2 + 4(-3)^3 + 4(-3)^4 + 4(-3)^5$
Step 2:
Calculate the value of the summation by simplifying each term.
$-732$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a sum of terms generated by substituting values into a given formula. In this case, the formula is $4(-3)^i$, where $i$ takes on integer values from 1 to 5.
To solve this problem, we follow these steps:
Expansion of the Series: We start by expanding the series, which means writing out each term of the summation explicitly. This is done by substituting the values of $i$ into the formula $4(-3)^i$.
Simplification: After expanding, we simplify the expression by performing the exponentiation and multiplication for each term. Since the base of the exponent is negative (-3), the sign of each term will alternate between positive and negative as the exponent $i$ increases.
In the context of mathematical notation:
The summation symbol $\sum$ is used to represent the sum of a sequence of terms.
The exponentiation $a^b$ means that the base $a$ is multiplied by itself $b$ times.
The sequence of terms generated by the formula $4(-3)^i$ is an example of a geometric series, where each term is a constant multiple of the previous term.
To correctly render the formula and data in LaTeX format, we use the caret symbol (^) for exponents and parentheses to ensure the correct order of operations. The LaTeX format is widely used in mathematical and scientific documents for its ability to clearly represent complex equations and expressions.
