To evaluate the summation of the expression 2n + 5 from n = 1 to n = 12, we follow these steps:
- Understanding the Summation Notation: The notation ∑ represents the sum of an expression evaluated at different values of n. In this case, we start at n = 1 and end at n = 12.
- Defining the Expression: The expression we want to sum is 2n + 5.
- Calculating Each Term: We will substitute each integer value of n from 1 to 12 into the expression:
- For n = 1: 2(1) + 5 = 2 + 5 = 7
- For n = 2: 2(2) + 5 = 4 + 5 = 9
- For n = 3: 2(3) + 5 = 6 + 5 = 11
- For n = 4: 2(4) + 5 = 8 + 5 = 13
- For n = 5: 2(5) + 5 = 10 + 5 = 15
- For n = 6: 2(6) + 5 = 12 + 5 = 17
- For n = 7: 2(7) + 5 = 14 + 5 = 19
- For n = 8: 2(8) + 5 = 16 + 5 = 21
- For n = 9: 2(9) + 5 = 18 + 5 = 23
- For n = 10: 2(10) + 5 = 20 + 5 = 25
- For n = 11: 2(11) + 5 = 22 + 5 = 27
- For n = 12: 2(12) + 5 = 24 + 5 = 29
- Summing the Values: Now we simply add all these computed values together:
So, summing it up:
- 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29
Let’s do the calculation step-by-step:
- 7 + 9 = 16
- 16 + 11 = 27
- 27 + 13 = 40
- 40 + 15 = 55
- 55 + 17 = 72
- 72 + 19 = 91
- 91 + 21 = 112
- 112 + 23 = 135
- 135 + 25 = 160
- 160 + 27 = 187
- 187 + 29 = 216
Thus, the final result of the summation of 2n + 5 from n = 1 to n = 12 is 216.