To find the sum of the finite arithmetic sequence defined by the expression 3n + 3 from n = 1 to n = 13, we first need to calculate the values of the expression for each integer n in this range.
The sequence generated by the expression for each n is:
- For n = 1: 3(1) + 3 = 6
- For n = 2: 3(2) + 3 = 9
- For n = 3: 3(3) + 3 = 12
- For n = 4: 3(4) + 3 = 15
- For n = 5: 3(5) + 3 = 18
- For n = 6: 3(6) + 3 = 21
- For n = 7: 3(7) + 3 = 24
- For n = 8: 3(8) + 3 = 27
- For n = 9: 3(9) + 3 = 30
- For n = 10: 3(10) + 3 = 33
- For n = 11: 3(11) + 3 = 36
- For n = 12: 3(12) + 3 = 39
- For n = 13: 3(13) + 3 = 42
Now, we will sum all these values:
6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30 + 33 + 36 + 39 + 42.
The sum can be calculated as follows:
- 6 + 9 = 15
- 15 + 12 = 27
- 27 + 15 = 42
- 42 + 18 = 60
- 60 + 21 = 81
- 81 + 24 = 105
- 105 + 27 = 132
- 132 + 30 = 162
- 162 + 33 = 195
- 195 + 36 = 231
- 231 + 39 = 270
- 270 + 42 = 312
Thus, the sum of the finite arithmetic sequence from n = 1 to n = 13 using the expression 3n + 3 is 312.