Which expression is equivalent to the given expression? 6 (2i – 3) i=-3
YOUR ANSWER CORRECT ANSWER
5 (2i – 3) i=-2
5 9+ (2–3) i=-2
9+ (2i – 3) i=-2
5
5 -9+ (2i – 3) ) i=-3
The Correct Answer and Explanation
6 (2i – 3) with i = -3
But this is ambiguous because it looks like summation notation or evaluation of an expression at a specific value of $i$. I assume the original question is about a summation or series:
Is the original expression:
$$
\sum_{i=-3}^6 (2i – 3)
$$
or something similar?
Your provided expression is unclear, so I’ll interpret it as:
Find an expression equivalent to:
$$
\sum_{i=-3}^6 (2i – 3)
$$
And your answer choices look like variants of sums with different indices or terms.
Step 1: Understand the sum notation
$$
\sum_{i=a}^b f(i)
$$
means sum $f(i)$ as $i$ goes from $a$ to $b$, including both $a$ and $b$.
Step 2: Calculate the sum $\sum_{i=-3}^6 (2i – 3)$
We want to find the sum of terms:
$$
(2(-3) – 3) + (2(-2) – 3) + \cdots + (2(6) – 3)
$$
First, find the number of terms:
From $-3$ to $6$, inclusive, there are $6 – (-3) + 1 = 6 + 3 + 1 = 10$ terms.
Step 3: Use formula for sum of linear sequence
Rewrite the sum:
$$
S = \sum_{i=-3}^6 (2i – 3) = 2 \sum_{i=-3}^6 i – 3 \sum_{i=-3}^6 1
$$
- $\sum_{i=-3}^6 i$ is the sum of integers from $-3$ to $6$.
- $\sum_{i=-3}^6 1$ counts the number of terms.
Number of terms: 10 (as above).
Calculate $\sum_{i=-3}^6 i$:
Sum of integers from $m$ to $n$ is:
$$
\frac{(n + m)(n – m + 1)}{2}
$$
Plug in $m = -3$, $n = 6$:
$$
\frac{(6 + (-3))(6 – (-3) + 1)}{2} = \frac{(3)(6 + 3 + 1)}{2} = \frac{3 \times 10}{2} = \frac{30}{2} = 15
$$
So:
$$
S = 2 \times 15 – 3 \times 10 = 30 – 30 = 0
$$
Step 4: Compare to answer choices
Your answer choices are:
- $5 \sum_{i=-2} (2i – 3)$
- $5 \cdot 9 + \sum_{i=-2} (2 – 3)$
- $\sum_{i=-2} (2i – 3)$
- $5$
- $5 – 9 + \sum_{i=-3} (2i – 3)$
Since the sum from $-3$ to 6 equals 0, let’s test which of these might equal 0.
- Option 3: $\sum_{i=-2} (2i – 3)$ — incomplete notation (no upper limit). Assuming it’s $\sum_{i=-2}^5 (2i – 3)$ or similar, this is different from the original sum and likely not equal to 0.
- Option 4: $5$ — this is just a number 5, which is not equal to 0.
- Option 5: $5 – 9 + \sum_{i=-3} (2i – 3)$ — again incomplete summation notation, but $5 – 9 = -4$, so this would depend on the sum.
Without clearer options, the only exact equivalent sum expression would be:
$$
\sum_{i=-3}^6 (2i – 3) = 0
$$
Summary:
- The sum $\sum_{i=-3}^6 (2i – 3)$ is equivalent to 0.
- This is found by separating the sum into two parts and calculating each with formulas for arithmetic sums.
- The answer choices provided are incomplete or unclear, but none seems to equal zero directly.
- The key is that summing a linear expression over a symmetric range around zero or near it can lead to zero because positive and negative terms cancel out.
Detailed Explanation (300 words)
The original expression $\sum_{i=-3}^6 (2i – 3)$ involves summing a linear function of $i$, specifically $2i – 3$, as $i$ goes from $-3$ to 6 inclusive. To find an equivalent expression or evaluate it, you can break the summation into simpler parts using properties of summations.
First, the summation can be split into the sum of two parts:
$$
\sum_{i=-3}^6 (2i – 3) = \sum_{i=-3}^6 2i – \sum_{i=-3}^6 3
$$
Because summation is linear, constants can be factored out:
$$
= 2 \sum_{i=-3}^6 i – 3 \sum_{i=-3}^6 1
$$
Next, evaluate each summation separately:
- $\sum_{i=-3}^6 i$ is the sum of integers from $-3$ up to $6$. Using the formula for the sum of an arithmetic sequence, which is $\frac{(n + m)(n – m + 1)}{2}$ where $m = -3$ and $n = 6$, you find the sum is 15.
- $\sum_{i=-3}^6 1$ simply counts the number of terms. From $-3$ to 6, there are 10 terms.
Plugging these values back in:
$$
2 \times 15 – 3 \times 10 = 30 – 30 = 0
$$
So the sum evaluates to zero.
This explains that the sum of a linear function over this symmetric range cancels out exactly.
Regarding the answer choices, without exact summation bounds or clarifications, none directly matches this calculation except the expression that fully captures the sum from $-3$ to $6$. Simplifying or shifting indices as in other choices will change the value, so the correct answer is the original summation $\sum_{i=-3}^6 (2i – 3)$, which equals zero.
