To find the sum of the first 30 terms of the sequence defined by the formula an = 6n + 5, we start by identifying how the terms are generated using this formula.
The first term corresponds to n = 1, the second term to n = 2, and so on. Thus, the first few terms can be calculated as follows:
- a1 = 6(1) + 5 = 11
- a2 = 6(2) + 5 = 17
- a3 = 6(3) + 5 = 23
- a4 = 6(4) + 5 = 29
- …
- a30 = 6(30) + 5 = 185
Now we can see that the sequence is an arithmetic progression where the first term a1 = 11 and the common difference d = a2 – a1 = 17 – 11 = 6.
The formula to find the sum Sn of the first n terms of an arithmetic sequence is:
Sn = (n/2) * (2a1 + (n – 1)d)
Substituting the values we have:
- n = 30
- a1 = 11
- d = 6
Now plug these into the formula:
S30 = (30 / 2) * (2 * 11 + (30 – 1) * 6)
S30 = 15 * (22 + 174)
S30 = 15 * 196
S30 = 2940
Therefore, the sum of the first 30 terms of the sequence is 2940.