Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 10 terms of the geometric sequence 4,12,36,108,324
To find the sum of the first 10 terms of the geometric sequence 4, 12, 36, 108, 324, we first need to determine the common ratio (r) of the sequence.
To do this, we divide each term by its preceding term to find the common ratio:
r = 12/4 = 3
r = 36/12 = 3
r = 108/36 = 3
r = 324/108 = 3
Since the common ratio is constant at 3, we can now calculate the sum of the first 10 terms using the formula for the sum of a geometric sequence:
S_n = a(1 - r^n) / (1 - r)
Where:
- S_n is the sum of the first n terms
- a is the first term of the sequence
- r is the common ratio
- n is the number of terms
Plugging in the values:
S_10 = 4(1 - 3^10) / (1 - 3)
S_10 = 4(1 - 59049) / -2
S_10 = 4(-59048) / -2
S_10 = -236192 / -2
S_10 = 118096
Therefore, the sum of the first 10 terms of the geometric sequence 4, 12, 36, 108, 324 is 118096.