Arithmetic vs Geometric Sequence Differences

You have a list of numbers and you need to know whether to add a constant or multiply by a constant to get the next term. That distinction, arithmetic vs geometric sequence, determines which formula you use and whether your series has a finite sum. An arithmetic sequence adds the same number each time; a geometric sequence multiplies by the same number each time. The multiplier is called the common ratio (r), and it must be non-zero. A sequence with r = 0 collapses to zero after the first term, which is still technically geometric but rarely useful. A sequence with r = 1 is a constant sequence, still geometric. The common ratio (r) defines the sequence's behavior: growth when |r| > 1, decay when 0 < |r| < 1, and alternation when r is negative.

An arithmetic sequence changes by a common difference (d), which is added to each term. A geometric sequence changes by a common ratio (r), which multiplies each term. For an arithmetic sequence, the graph of terms is a straight line, linear growth. For a geometric sequence, the graph is a curve that bends upward or downward, exponential growth. The formulas reflect this. The n-th term of an arithmetic sequence is aₙ = a₁ + (n-1)d. The n-th term of a geometric sequence is aₙ = a₁ * r^(n-1), from OpenStax Algebra and Trigonometry 2e, section 13.3. The partial sum of an arithmetic series is Sₙ = n/2 * (2a₁ + (n-1)d). The partial sum of a geometric series is Sₙ = a₁(1 - rⁿ)/(1 - r) for r ≠ 1, from OpenStax Algebra and Trigonometry 2e, section 13.4. For r = 1, the partial sum formula becomes Sₙ = n*a₁, because the standard formula is undefined at r = 1.

To determine whether a sequence is arithmetic or geometric, compute the difference between consecutive terms. If the difference is constant, the sequence is arithmetic. Then compute the ratio of consecutive terms. If the ratio is constant, the sequence is geometric. A sequence can be neither. For example, the sequence 2, 5, 10, 17 has differences 3, 5, 7 (not constant) and ratios 2.5, 2, 1.7 (not constant). That sequence is neither arithmetic nor geometric. When testing, check the ratio from two non-consecutive terms using r = (aₙ / aₘ)^(1/(n-m)). If the ratio of two given terms is negative and the root index (n - m) is even, the sign of r is ambiguous; you must check the sign pattern of the sequence to decide. A failure mode here is taking the wrong root, which loses the sign of r.

The explicit formula for an arithmetic sequence is aₙ = a₁ + (n-1)d. The explicit formula for a geometric sequence is aₙ = a₁ * r^(n-1). The recursive form of an arithmetic sequence is aₙ = aₙ₋₁ + d. The recursive form of a geometric sequence is aₙ = r * aₙ₋₁. When you are given two terms, finding the common difference in an arithmetic sequence is straightforward: d = (aₙ - aₘ)/(n - m). Finding the common ratio in a geometric sequence uses r = (aₙ / aₘ)^(1/(n-m)). The first term is then a₁ = aₙ / r^(n-1). For a geometric series, the infinite sum formula S∞ = a₁/(1 - r) works only when |r| < 1, the convergence condition from Stewart Calculus, section 11.2. If |r| ≥ 1, the series diverges and has no finite sum. A common failure mode is using the infinite sum formula when |r| ≥ 1, producing a finite number for a divergent series.

Arithmetic sequences produce linear growth. The graph of terms against index is a straight line with slope equal to the common difference. Geometric sequences produce exponential growth when |r| > 1, and exponential decay when 0 < |r| < 1. The graph of a geometric sequence with r > 1 curves upward, increasing faster as n increases. With r between 0 and 1, the graph curves downward, approaching zero. With r negative, the terms alternate sign, and the graph oscillates. This visual difference helps you identify a sequence type at a glance: if the points form a line, it is arithmetic; if they curve, it is geometric.

Not every sequence is arithmetic or geometric. A sequence like 1, 4, 9, 16 has differences 3, 5, 7 (not constant) and ratios 4, 2.25, 1.777... (not constant). That sequence is a list of perfect squares, which follows a quadratic pattern. Another example is the Fibonacci sequence: 1, 1, 2, 3, 5, 8. The differences are 0, 1, 1, 2, 3 (not constant), and the ratios are 1, 2, 1.5, 1.666..., 1.6 (not constant). It is neither arithmetic nor geometric. When you encounter a sequence that does not have a constant difference or a constant ratio, look for other patterns: squares, cubes, prime numbers, or recursive definitions. A geometric mean relates to the middle term of a three-term geometric sequence. The middle term equals the square root of the product of its neighbors.

Example 1: Determine if the sequence 3, 6, 12, 24 is arithmetic or geometric. The differences are 3, 6, 12 (not constant). The ratios are 2, 2, 2 (constant). It is geometric with common ratio r = 2.

Example 2: Determine if the sequence 10, 7, 4, 1 is arithmetic or geometric. The differences are -3, -3, -3 (constant). The ratios are 0.7, 0.571, 0.25 (not constant). It is arithmetic with common difference d = -3.

Example 3: Find the common ratio for a geometric sequence where a₃ = 12 and a₆ = 96. Use r = (aₙ / aₘ)^(1/(n-m)) = (96 / 12)^(1/(6-3)) = 8^(1/3) = 2. The common ratio is 2.

Example 4: Find the first term of a geometric sequence where a₄ = 40 and r = 2. Use a₁ = aₙ / r^(n-1) = 40 / 2^(4-1) = 40 / 8 = 5. The first term is 5.

Example 5: Find the partial sum of the first 5 terms of a geometric sequence with a₁ = 3 and r = 2. Use Sₙ = a₁(1 - rⁿ)/(1 - r) = 3(1 - 2⁵)/(1 - 2) = 3(1 - 32)/(-1) = 3(-31)/(-1) = 93. The partial sum S₅ is 93.

Example 6: Determine if the infinite geometric series with a₁ = 5 and r = 0.5 converges. Since |r| = 0.5 < 1, it converges. The infinite sum is S∞ = a₁/(1 - r) = 5/(1 - 0.5) = 5/0.5 = 10.

To tell the two sequence types apart, run two tests. Test 1: subtract consecutive terms. If the difference is constant, it is arithmetic. Test 2: divide consecutive terms. If the ratio is constant, it is geometric. If neither test passes, the sequence is something else. The failure mode most people hit is using the arithmetic formula aₙ = a₁ + (n-1)d on a geometric sequence. That produces wrong terms from the start. Another failure mode is misidentifying the first term when the sequence index starts at 0 instead of 1, causing an off-by-one error in the exponent. Check the index carefully.

The common difference (d) is the constant added to each term in an arithmetic sequence. It tells you the slope of the line on a graph. The common ratio (r) is the constant multiplier in a geometric sequence. It tells you whether the sequence grows, decays, or alternates. When |r| > 1, terms grow without bound; when |r| < 1, terms approach zero; when r is negative, the terms alternate sign. The common ratio also determines convergence of the infinite series: if |r| < 1, the series converges; if |r| ≥ 1, it diverges. A common confusion is between r and |r| for convergence. A series with r = -2 has |r| = 2, which is ≥ 1, so it diverges, even though r < 1.

This material suits Algebra 2 and precalculus students who need to find terms, the common ratio, or the number of terms from given information. It also suits calculus students who need to determine whether an infinite geometric series converges and, if so, compute its sum. Teachers who need a reliable answer key or a way to demonstrate the relationship between the sequence formula and the series sum formula will also find it useful. Skip this if you need to compute the sum of an arithmetic series, use an arithmetic series calculator instead. Also skip if you need the sum of a power series beyond the geometric form, use a power series convergence calculator. Students looking for proofs of convergence tests other than the ratio test for geometric series should consult a calculus textbook section on the ratio test.