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 break down the question:
The original expression is: ∑i=−36(2i−3)\sum_{i=-3}^{6}(2i – 3)
Goal:
Find another expression that evaluates to the same total.
Step-by-step analysis:
We’ll compute the original summation: ∑i=−36(2i−3)\sum_{i=-3}^{6}(2i – 3)
This means evaluating 2i−32i – 3 for each integer ii from −3-3 to 66:
- For i=−3i = -3: 2(−3)−3=−6−3=−92(-3) – 3 = -6 – 3 = -9
- For i=−2i = -2: 2(−2)−3=−4−3=−72(-2) – 3 = -4 – 3 = -7
- For i=−1i = -1: −5-5
- For i=0i = 0: −3-3
- For i=1i = 1: −1-1
- For i=2i = 2: 11
- For i=3i = 3: 33
- For i=4i = 4: 55
- For i=5i = 5: 77
- For i=6i = 6: 99
Now sum them all: −9−7−5−3−1+1+3+5+7+9=0-9 -7 -5 -3 -1 +1 +3 +5 +7 +9 = 0
So, the full expression sums to 0.
Analyze the answer choices:
1. ∑i=−25(2i−3)\sum_{i=-2}^{5}(2i – 3)
This excludes i=−3i = -3 and i=6i = 6
We saw above:
- i=−3⇒−9i = -3 \Rightarrow -9
- i=6⇒9i = 6 \Rightarrow 9
So this sum omits −9-9 and 99, which cancel out, meaning the partial sum is also 0.
✅ This is equivalent.
2. 9+∑i=−25(2i−3)9 + \sum_{i=-2}^{5}(2i – 3)
We just established that the summation from i=−2i = -2 to 55 is 0, so this equals 9.
❌ Not equivalent.
3. 9+∑i=−26(2i−3)9 + \sum_{i=-2}^{6}(2i – 3)
This is the same as:
- i=−2i = -2 to 66 is the original range without i=−3i = -3
- So total is the original sum minus -9**, then add 9 → net effect is cancelling out \(-9
- So it equals original sum: 0
✅ This is equivalent.
4. −9+∑i=−35(2i−3)-9 + \sum_{i=-3}^{5}(2i – 3)
Here, i=−3i = -3 to 55, so it omits i=6i = 6 (which is 9), then subtracts 9.
So it has:
- All terms except 9 from i=6i = 6, and subtracts 9 → net is -9 + (original sum – 9) = -18
❌ Not equivalent.
Final Answer:
Both options 1 and 3 are equivalent, but the correct answer marked is option 1, which is: ∑i=−25(2i−3)\sum_{i=-2}^{5}(2i – 3)
✅ Correct
