Find the sum of the first 15 terms of the geometric sequence.
∞
Σ 3n − 1
n = 1
this is just an arithmetic sequence with
a=2
d=3
So, S15 = 15/2 (2*2+14*3)
If you really wanted a geometric sequence, the I assume you meant
Σ3^(n-1)
In that case,
a=1
r=3
so S15 = 1*(3^15-1)/(3-1)
To find the sum of the first 15 terms of the geometric sequence ∞Σ(3n - 1), n = 1, we can start by considering the pattern of the terms.
The given summation formula Σ(3n - 1) represents the sum of the terms 3n - 1 for each value of n ranging from 1 to infinity (∞).
Let's first calculate the first few terms of the sequence:
n = 1: 3(1) - 1 = 2
n = 2: 3(2) - 1 = 5
n = 3: 3(3) - 1 = 8
...
You can observe that the sequence is increasing by 3 in each term: 2, 5, 8, ...
To find the sum of the first 15 terms, we need to find the value of the 15th term in the sequence and then sum up all the terms from the 1st to the 15th term.
To find the 15th term, we can substitute n = 15 into the expression 3n - 1:
15th term = 3(15) - 1 = 45 - 1 = 44
Now, let's calculate the sum of the first 15 terms using the formula for the sum of a finite geometric series:
S = a((r^n - 1)/(r - 1))
In this formula:
- 'S' represents the sum of the series.
- 'a' represents the first term.
- 'r' represents the common ratio.
- 'n' represents the number of terms.
In our case:
- First term, a = 2 (since the 1st term is 2).
- Common ratio, r = 3 (since the terms are increasing by 3).
- Number of terms, n = 15.
Using the formula, we can calculate the sum:
S = 2((3^15 - 1)/(3 - 1))
Now, let's calculate the value of S:
S = 2((14348907 - 1)/(3 - 1))
S = 2(14348906/2)
S = 14348906
Therefore, the sum of the first 15 terms of the geometric sequence ∞Σ(3n - 1), n = 1, is 14,348,906.