How do you find the 12th partial sum of the series involving the summation of -2i – 10 from i = 1 to infinity?

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To find the 12th partial sum of the summation of the series defined by the formula -2i – 10 for i = 1 to 12, you need to follow these steps:

  1. Understand the Formula: The general term of the series is given by -2i – 10. This means that for each integer value of i, you will calculate the term by substituting the value of i into the formula.
  2. Calculate Individual Terms: You will calculate the individual terms for i = 1 through i = 12:
    • For i = 1: -2(1) – 10 = -2 – 10 = -12
    • For i = 2: -2(2) – 10 = -4 – 10 = -14
    • For i = 3: -2(3) – 10 = -6 – 10 = -16
    • For i = 4: -2(4) – 10 = -8 – 10 = -18
    • For i = 5: -2(5) – 10 = -10 – 10 = -20
    • For i = 6: -2(6) – 10 = -12 – 10 = -22
    • For i = 7: -2(7) – 10 = -14 – 10 = -24
    • For i = 8: -2(8) – 10 = -16 – 10 = -26
    • For i = 9: -2(9) – 10 = -18 – 10 = -28
    • For i = 10: -2(10) – 10 = -20 – 10 = -30
    • For i = 11: -2(11) – 10 = -22 – 10 = -32
    • For i = 12: -2(12) – 10 = -24 – 10 = -34
  3. Sum the Terms: Now, you will sum all the values obtained from the terms calculated above:
S = -12 + (-14) + (-16) + (-18) + (-20) + (-22) + (-24) + (-26) + (-28) + (-30) + (-32) + (-34)

Calculating this sum:

S = -12 - 14 - 16 - 18 - 20 - 22 - 24 - 26 - 28 - 30 - 32 - 34

Simplifying:
S = -12 – 14 = -26
S = -26 – 16 = -42
S = -42 – 18 = -60
S = -60 – 20 = -80
S = -80 – 22 = -102
S = -102 – 24 = -126
S = -126 – 26 = -152
S = -152 – 28 = -180
S = -180 – 30 = -210
S = -210 – 32 = -242
S = -242 – 34 = -276

Thus, the 12th partial sum of the series is -276.

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