Geometric Series: Sum of the First Five Terms of 7, 14, 28, 56, 112
Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Finding the Common Ratio
Given the sequence 7, 14, 28, 56, 112, the common ratio (r) can be found by dividing any term by the preceding term. For example:
(r) 14 / 7 2 (r) 28 / 14 2 (r) 56 / 28 2 (r) 112 / 56 2The common ratio is therefore 2.
Sum of the First Five Terms
The sum of the first five terms of a geometric sequence can be found using the formula:
Sn a(rn - 1) / (r - 1)
Where a is the first term, r is the common ratio, and n is the number of terms.
For the sequence 7, 14, 28, 56, 112:
a 7 r 2 n 5Substituting these values into the formula:
S5 7(25 - 1) / (2 - 1) 7(32 - 1) / 1 7 * 31 217
The sum of the first five terms is 217.
Verification
The first five terms are 7, 14, 28, 56, and 112. Adding these together:
7 14 21 21 28 49 49 56 105 105 112 217Thus, the sum of the first five terms is indeed 217.
Conclusion
The sum of the first five terms of the geometric sequence 7, 14, 28, 56, 112 is 217.
Additional Resources
For further reading on geometric series, you can refer to the following resources:
Geometric Progression - Wikipedia