Evaluate the Summation of -5n – 1 from n = 3 to 12

To evaluate the summation of the expression -5n – 1 from n = 3 to n = 12, we will proceed by calculating the value of the expression for each integer n in this range and then summing those values.

The summation can be expressed mathematically as:

S = Σ (-5n – 1) from n = 3 to 12

Now, let’s calculate it step by step:

1. First, we identify the range of n: from 3 to 12. This includes the values: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
2. Next, we compute the value of -5n – 1 for each of these values:
• For n = 3: -5(3) – 1 = -15 – 1 = -16
• For n = 4: -5(4) – 1 = -20 – 1 = -21
• For n = 5: -5(5) – 1 = -25 – 1 = -26
• For n = 6: -5(6) – 1 = -30 – 1 = -31
• For n = 7: -5(7) – 1 = -35 – 1 = -36
• For n = 8: -5(8) – 1 = -40 – 1 = -41
• For n = 9: -5(9) – 1 = -45 – 1 = -46
• For n = 10: -5(10) – 1 = -50 – 1 = -51
• For n = 11: -5(11) – 1 = -55 – 1 = -56
• For n = 12: -5(12) – 1 = -60 – 1 = -61
3. Now, we have the values:
• -16, -21, -26, -31, -36, -41, -46, -51, -56, -61
4. Finally, we sum these values:

S = -16 – 21 – 26 – 31 – 36 – 41 – 46 – 51 – 56 – 61

Calculating the sum step by step, we get:

• -16 – 21 = -37
• -37 – 26 = -63
• -63 – 31 = -94
• -94 – 36 = -130
• -130 – 41 = -171
• -171 – 46 = -217
• -217 – 51 = -268
• -268 – 56 = -324
• -324 – 61 = -385

The final result for the summation is:

Σ (-5n – 1) from n = 3 to 12 = -385