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Evaluate: sum_(n=0)^(4)(-5x-5n) Answer:

Evaluate:\newline∑n=04(−5x−5n) \sum_{n=0}^{4}(-5 x-5 n) \newlineAnswer:

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Full solution

Q. Evaluate:\newline∑n=04(−5x−5n) \sum_{n=0}^{4}(-5 x-5 n) \newlineAnswer:
  1. Write Series to Sum: Write down the series that needs to be summed.\newlineThe series is given by ∑n=04(−5x−5n)\sum_{n=0}^{4}(-5x - 5n).
  2. Evaluate Series Terms: Evaluate the series by plugging in the values of nn from 00 to 44 and adding the terms.\newline∑n=04(−5x−5n)=(−5x−5⋅0)+(−5x−5⋅1)+(−5x−5⋅2)+(−5x−5⋅3)+(−5x−5⋅4)\sum_{n=0}^{4}(-5x - 5n) = (-5x - 5\cdot 0) + (-5x - 5\cdot 1) + (-5x - 5\cdot 2) + (-5x - 5\cdot 3) + (-5x - 5\cdot 4)
  3. Simplify Expression: Simplify the expression by performing the multiplication and addition.\newline=(−5x−0)+(−5x−5)+(−5x−10)+(−5x−15)+(−5x−20)= (-5x - 0) + (-5x - 5) + (-5x - 10) + (-5x - 15) + (-5x - 20)\newline=−5x+−5x+−5x+−5x+−5x−0−5−10−15−20= -5x + -5x + -5x + -5x + -5x - 0 - 5 - 10 - 15 - 20
  4. Combine Like Terms: Combine like terms.\newline=−5x×5−(0+5+10+15+20)= -5x \times 5 - (0 + 5 + 10 + 15 + 20)\newline=−25x−(0+5+10+15+20)= -25x - (0 + 5 + 10 + 15 + 20)
  5. Calculate Sum of Constants: Calculate the sum of the constants. 0+5+10+15+20=500 + 5 + 10 + 15 + 20 = 50
  6. Substitute Constants: Substitute the sum of the constants back into the expression.\newline=−25x−50= -25x - 50
  7. Final Simplified Expression: The final simplified expression represents the sum of the series. The sum of the series from n=0n=0 to n=4n=4 is −25x−50-25x - 50.

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