Final answer:
The value of the alternating series 4Σn=1(-1)ⁿ(3n+2) is calculated by expanding the series, summing the terms with their respective signs, and then multiplying the sum by 4, leading to a final value of 24.
Step-by-step explanation:
The question asks for the value of a mathematical series, to be precise, a finite alternating series. To find the value of 4Σn=1(-1)ⁿ(3n+2), first expand the series and then apply the summation and alternating signs accordingly.
Here's the series expansion:
When n=1, the term is (-1)¹(3×1+2) = -5
Add these terms together: -5 + 8 - 11 + 14 = 6.
Finally, multiply the sum by 4 as indicated by the problem: 4 * 6 = 24.
The value of 4Σn=1(-1)ⁿ(3n+2) is 24.
By expanding the alternating series and summing up the terms, taking into consideration the alternating signs and the sequence of n values, we arrived at a sum of 6. Multiplying this sum by 4 as the series expression requires, we obtain the final value of the series, which is 24.