Evaluate the Summation sum from i=3 to 7 of -(-2/3)^(i-1)
The problem you have presented is an exercise in evaluating a finite mathematical series. Specifically, it asks you to compute the sum of a series where the general term is given by the expression -(-2/3)^(i-1). The summation should be carried out for the integer values of the index variable i, starting at i=3 and ending at i=7. This requires you to substitute the values 3, 4, 5, 6, and 7 into the expression, calculate each term, and then sum all the resulting terms together to find the final answer.
$\sum_{i = 3}^{7} - \left(\left(\right. - \frac{2}{3} \left.\right)\right)^{i - 1}$
Answer
Solution:
Step:1 Firstly, write out the terms of the summation individually for each value of $i$ from 3 to 7. The terms are $- \left(-\frac{2}{3}\right)^{3-1}, - \left(-\frac{2}{3}\right)^{4-1}, - \left(-\frac{2}{3}\right)^{5-1}, \ldots, - \left(-\frac{2}{3}\right)^{7-1}$.
Step:2 Next, calculate the value of each term and sum them up. The result is $- \frac{220}{729}$.
Step:3 Finally, present the sum in its exact form and its decimal approximation. The exact value is $- \frac{220}{729}$, and the decimal equivalent is approximately $- 0.30178326 \ldots$.
Knowledge Notes:
The problem involves evaluating a finite geometric series. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The summation in question can be written as:
$$\sum_{i=3}^{7} -\left(-\frac{2}{3}\right)^{i-1}$$
To solve this, you need to understand the following concepts:
Summation Notation: This is a way to represent the sum of a sequence of terms. The notation $\sum$ indicates the sum, the expression to the right of the summation symbol indicates the term to be added, and the values below and above the symbol indicate the starting and ending indices, respectively.
Geometric Series: A geometric series is a series with a constant ratio between successive terms. The general form of a geometric series is $a, ar, ar^2, ar^3, \ldots$ where $a$ is the first term and $r$ is the common ratio.
Exponents: When raising a negative number to an even power, the result is positive; when raising it to an odd power, the result is negative.
Exact and Decimal Form: The exact form of a number is the precise value, often represented as a fraction. The decimal form is an approximation of the exact form, expressed in base 10 with a finite or infinite number of decimal places.
In this problem, the common ratio is $-\frac{2}{3}$, and we are looking at the series' terms from the 3rd to the 7th term. Each term is found by raising the common ratio to the power of $i-1$ and then taking the negative of that result. After expanding and simplifying the series, the sum is presented in both exact and decimal forms.
