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:
It looks like you’re trying to find an expression equivalent to the sum: 6∑i=−3(2i−3)6 \sum_{i=-3} (2i – 3)
But your message has some formatting issues, so I’ll clarify what I think you’re asking and then explain step-by-step.
Step 1: Understand the original expression
Your original expression looks like: 6∑i=−3(2i−3)6 \sum_{i=-3} (2i – 3)
But it’s incomplete because a summation needs both lower and upper limits for ii. Usually, a summation is written as: ∑i=abf(i)\sum_{i=a}^{b} f(i)
where aa is the lower limit and bb is the upper limit.
Since only the lower limit i=−3i = -3 is given, I assume the summation is over some range, maybe i=−3i = -3 to some upper value, or maybe just the single term when i=−3i = -3.
Step 2: Check the possible interpretations
Interpretation A: Single term at i=−3i=-3 6(2(−3)−3)=6(−6−3)=6(−9)=−546(2(-3) – 3) = 6(-6 – 3) = 6(-9) = -54
So the expression evaluates to −54-54.
Interpretation B: Summation from i=−3i=-3 to some upper limit
If the upper limit is missing, but the question gives alternative expressions involving summations from i=−2i=-2, etc., maybe the sum goes from −3-3 to −1-1 or from −3-3 to −2-2.
Step 3: Verify the alternative expressions given
The options you wrote are unclear, but they seem to try to rewrite the sum with different limits or coefficients.
Let’s try to test the sum for a small range and see if the expressions are equivalent.
For example, calculate: 6∑i=−3−1(2i−3)6 \sum_{i=-3}^{-1} (2i – 3)
Calculate the sum inside first:
- When i=−3i=-3: 2(−3)−3=−6−3=−92(-3) – 3 = -6 – 3 = -9
- When i=−2i=-2: 2(−2)−3=−4−3=−72(-2) – 3 = -4 – 3 = -7
- When i=−1i=-1: 2(−1)−3=−2−3=−52(-1) – 3 = -2 – 3 = -5
Sum = −9+(−7)+(−5)=−21-9 + (-7) + (-5) = -21
Multiply by 6: 6×(−21)=−1266 \times (-21) = -126
Now, check a similar sum with 5∑i=−2(2i−3)5 \sum_{i=-2} (2i – 3):
Assuming the sum runs from i=−2i=-2 to −1-1:
- i=−2i=-2: −7-7
- i=−1i=-1: −5-5
Sum = −7+(−5)=−12-7 + (-5) = -12
Multiply by 5: 5×(−12)=−605 \times (-12) = -60
Not equal to −126-126.
Step 4: Conclusion
Without clear summation bounds, it’s tricky to give a definitive equivalent expression. But based on standard summation properties:
- You can factor constants out of the sum.
- You can change the index of summation by shifting indices and adjusting the sum limits accordingly.
- Multiplying the whole sum by a constant multiplies the result by that constant.
Final answer:
If your original expression is: 6∑i=−3n(2i−3)6 \sum_{i=-3}^n (2i – 3)
Then an equivalent expression with a shifted index j=i+1j = i + 1 (so i=j−1i = j – 1) can be: 6∑j=−2n+1(2(j−1)−3)=6∑j=−2n+1(2j−2−3)=6∑j=−2n+1(2j−5)6 \sum_{j=-2}^{n+1} \bigl(2(j-1) – 3\bigr) = 6 \sum_{j=-2}^{n+1} (2j – 2 – 3) = 6 \sum_{j=-2}^{n+1} (2j – 5)
This is just a reindexing.
The expression involves a summation and multiplication by a constant. A summation like ∑i=ab(2i−3)\sum_{i=a}^b (2i – 3) sums the linear expression 2i−32i – 3 over integer values of ii from aa to bb. Multiplying the sum by a constant outside the summation is equivalent to multiplying each term inside the sum by that constant. Changing the lower or upper limits of the summation changes the terms included, so any equivalent expression must adjust for those differences. Also, reindexing the summation (shifting the index ii to j=i+kj = i + k) changes the limits and the expression inside the sum accordingly but keeps the overall value the same.
Without clear limits, the exact equivalent form cannot be determined. However, typical equivalent expressions involve either changing the index and limits of the sum or factoring out constants. The terms in the sum must correspond exactly for the sums to be equal.
If you provide the exact full summation limits and all answer choices clearly, I can confirm the correct equivalent expression precisely. Right now, the most correct answer is to clarify the sum limits and understand summation properties.
