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 is:

Let’s carefully analyze the problem you gave:

Original expression:
$6 \sum_{i=-3}^{?} (2i – 3)$

But the problem statement seems a bit jumbled or incomplete — it shows:
6 (2i – 3) i=-3

This looks like a summation notation issue or missing parts. Based on the options, it seems like the question wants to know which expression is equivalent to the original summation:

$$
6 \sum_{i=-3}^{something} (2i – 3)
$$

and the options involve expressions like:

$5 \sum_{i=-2} (2i – 3)$
$5 \sum_{i=-2} 9 + (2 – 3)$
$\sum_{i=-2} 9 + (2i – 3)$
$5$
$5 \sum_{i=-3} -9 + (2i – 3)$

Step 1: Clarify the original expression

Let’s assume the original expression is:

$$
6 \sum_{i=-3}^{n} (2i – 3)
$$

for some $n$, and we want to find an equivalent expression.

Step 2: Understand the summation of $2i – 3$

The sum from $i = a$ to $i = b$ of $2i – 3$ can be computed as:

$$
\sum_{i=a}^b (2i – 3) = 2 \sum_{i=a}^b i – 3 \sum_{i=a}^b 1 = 2 \left(\frac{(b+a)(b – a + 1)}{2}\right) – 3 (b – a + 1)
$$

Here:

$\sum_{i=a}^b i = \frac{(b+a)(b – a + 1)}{2}$ is the formula for the sum of an arithmetic series.
$\sum_{i=a}^b 1 = (b – a + 1)$ is the number of terms.

Step 3: Apply this for given limits

For example, if we consider the sum from $i = -3$ to $i = -3$ (only one term), then:

$$
\sum_{i=-3}^{-3} (2i – 3) = 2(-3) – 3 = -6 – 3 = -9
$$

If the sum goes from $i = -3$ to $i = -2$ (two terms):

$$
\sum_{i=-3}^{-2} (2i – 3) = (2(-3) – 3) + (2(-2) – 3) = (-6 – 3) + (-4 – 3) = -9 + (-7) = -16
$$

Step 4: Compare the given options

The option $5 \sum_{i=-2} (2i – 3)$ is incomplete (sum to what?).
Another option shows $9 + (2 – 3)$, which looks unrelated.
$\sum_{i=-2} 9 + (2i – 3)$ is not standard notation.
$5$ alone is a number, unlikely equivalent to a sum expression.
$5 \sum_{i=-3} -9 + (2i – 3)$ again is ambiguous.

Step 5: Make sense of the problem

If the original expression is:

$$
6 \sum_{i=-3}^{-3} (2i – 3)
$$

which equals

$$
6 \times (-9) = -54
$$

And if we consider

$$
5 \sum_{i=-2}^{-2} (2i – 3)
$$

this equals:

$$
5 \times (2(-2) – 3) = 5 \times (-4 – 3) = 5 \times (-7) = -35
$$

This is not equal to the original value.

Conclusion:

The original question is ambiguous because of missing limits or summation signs.

If I had to guess the intended question:

“Which expression is equivalent to $6 \sum_{i=-3}^k (2i – 3)$ for some $k$?” and the correct answer involves shifting the index or adjusting the multiplier.

Explanation of Equivalent Summation Expressions (300+ words):

Summation expressions can sometimes be rewritten in different but equivalent forms by changing the index of summation or adjusting the constant multipliers outside the summation.

For example, the summation

$$
\sum_{i=a}^b f(i)
$$

means adding the values of $f(i)$ starting at $i = a$ up to $i = b$. If we multiply the entire summation by a constant $c$, it’s the same as:

$$
c \sum_{i=a}^b f(i) = \sum_{i=a}^b c \times f(i)
$$

Also, sometimes, the index of summation $i$ can be shifted by an integer $m$ to rewrite the sum, such as:

$$
\sum_{i=a}^b f(i) = \sum_{j = a + m}^{b + m} f(j – m)
$$

where $j = i + m$. This change can simplify calculations or match given expressions.

In your problem, the expression involves $6 \sum (2i – 3)$, which could be rewritten with a different index starting at $i = -2$ instead of $i = -3$, but then the multiplier and limits would adjust accordingly to keep the sum equivalent.

When trying to find an equivalent expression, you need