Common Difference And Common Ratio Calculator

Arithmetic and Geometric Sequence Calculations:

\[ d = a_2 - a_1 \quad \text{(Arithmetic)} \] \[ r = \frac{a_2}{a_1} \quad \text{(Geometric)} \]

Result:

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1. What Are Common Difference and Common Ratio?

The common difference (d) is the constant difference between consecutive terms in an arithmetic sequence. The common ratio (r) is the constant ratio between consecutive terms in a geometric sequence.

2. How Does the Calculator Work?

The calculator uses these fundamental formulas:

\[ d = a_2 - a_1 \quad \text{(Arithmetic Sequence)} \] \[ r = \frac{a_2}{a_1} \quad \text{(Geometric Sequence)} \]

Where:

\( a_1 \) — First term in the sequence
\( a_2 \) — Second term in the sequence
\( d \) — Common difference (arithmetic)
\( r \) — Common ratio (geometric)

3. Importance in Sequences

Details: Identifying the common difference or ratio allows you to predict future terms in the sequence and understand its behavior. Arithmetic sequences grow linearly while geometric sequences grow exponentially.

4. Using the Calculator

Tips: Enter any two consecutive terms of your sequence and select whether it's arithmetic or geometric. The calculator will determine the common difference or ratio.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between arithmetic and geometric sequences?
A: Arithmetic sequences add a constant value each step, while geometric sequences multiply by a constant value.

Q2: Can the common ratio be negative?
A: Yes, a geometric sequence can have a negative common ratio, which makes the terms alternate in sign.

Q3: What if my sequence isn't arithmetic or geometric?
A: This calculator only works for arithmetic and geometric sequences. Other patterns require different approaches.

Q4: Can the common difference be zero?
A: Yes, but this would mean all terms in the arithmetic sequence are identical.

Q5: What if the first term is zero in a geometric sequence?
A: The common ratio would be undefined if a₁ is zero, as division by zero is impossible.