nth Term
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Geometric Sequence Calculator
Find nth terms and sums with a common ratio.
Sum of First n Terms
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Geometric Sequence Calculator: Growth by a Constant Ratio
A geometric sequence changes by multiplying by a constant ratio each step. That makes it the natural opposite of an arithmetic sequence. The calculator is useful whenever growth or decay happens by repeated multiplication rather than repeated addition.
Compound interest, bacteria growth, and depreciation are all geometric patterns in disguise. If the ratio is greater than 1, the sequence grows; if it is between 0 and 1, the sequence shrinks. The calculator shows both the nth term and the accumulated sum when that sum is defined. When paired x-y measurements look more linear than multiplicative, a linear regression calculator is often the better summary tool, and if you are undoing a straight-line map algebraically, an inverse function calculator reinforces the same reversible reasoning in function notation.
How the Formula Works
The nth term is the first term multiplied by the ratio raised to a power. The sum of the first n terms has a separate formula, and it only works cleanly when the ratio is not 1. That distinction is important because geometric sequences can grow very fast or shrink very fast depending on the ratio.
The calculator is helpful because it shows both outputs together. That makes it easy to inspect a growth pattern, check a homework answer, or compare scenarios where the ratio changes.
Ratio Greater Than 1
The sequence grows rapidly.
Ratio Between 0 and 1
The sequence shrinks toward zero.
That makes geometric sequences a powerful model for compounding and decay.
Real-World Use Case: Interest, Decay, and Scale
A sequence starting at 2 with ratio 3 becomes 2, 6, 18, 54, and so on. The 5th term is 162 and the sum of the first five terms is 242. If the ratio changes from 3 to 2, the numbers stay much smaller.
Because each step multiplies, the values can get large quickly. That is why a geometric calculator is useful for spotting growth before it gets out of hand. It is also a good fit for depreciation and decay where values shrink instead of expand.
In short, it is a compact tool for repeated multiplication.
Common Sequence Mistakes
First: confusing ratio with difference.
Second: forgetting that the sum formula changes when the ratio is 1.
Third: assuming every geometric sequence grows instead of some shrinking.
Once the ratio is clear, the pattern is easy to follow.
Reference Data Table
| a1 | r | n |
|---|---|---|
| 2 | 3 | 5 |
| 10 | 1/2 | 4 |
| 5 | 4 | 3 |
These examples show both growth and decay in one place.
Frequently Asked Questions
What makes it geometric?
Each term is multiplied by the same ratio.
What if the ratio is 1?
Then every term is the same.
Can it show sums?
Yes, for the first n terms.
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