To find the sum of the first 15 terms of the sequence defined by an = 10n + 21, we first need to identify the first 15 terms of the sequence.
We can calculate the first 15 terms by substituting values of n from 1 to 15:
- a1 = 10(1) + 21 = 31
- a2 = 10(2) + 21 = 41
- a3 = 10(3) + 21 = 51
- a4 = 10(4) + 21 = 61
- a5 = 10(5) + 21 = 71
- a6 = 10(6) + 21 = 81
- a7 = 10(7) + 21 = 91
- a8 = 10(8) + 21 = 101
- a9 = 10(9) + 21 = 111
- a10 = 10(10) + 21 = 121
- a11 = 10(11) + 21 = 131
- a12 = 10(12) + 21 = 141
- a13 = 10(13) + 21 = 151
- a14 = 10(14) + 21 = 161
- a15 = 10(15) + 21 = 171
Now we need to sum these values:
Sum = a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 + a11 + a12 + a13 + a14 + a15
Calculating this gives us:
Sum = 31 + 41 + 51 + 61 + 71 + 81 + 91 + 101 + 111 + 121 + 131 + 141 + 151 + 161 + 171
Using the formula for the sum of an arithmetic series, where Sn = n/2 * (first term + last term), we identify:
- n = 15
- first term = 31
- last term = 171
Substituting into the formula, we have:
S15 = 15/2 * (31 + 171) = 15/2 * 202 = 15 * 101 = 1515
Thus, the sum of the first 15 terms of the sequence is 1515.