Geometric Sequence & Series Calculator

Geometric Sequences Guide

What do I need to know about Geometric Sequence Formulas?

A geometric sequence multiplies by a constant ratio r each time: a, ar, ar², ar³... nth term: aₙ = a × r^(n−1). Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1, or Sₙ = na for r = 1. Sum to infinity (|r| < 1): S∞ = a/(1 − r). Example: 3, 6, 12, 24... (a = 3, r = 2). 8th term = 3 × 2⁷ = 384. Sum of 8 terms = 3(2⁸ − 1)/(2 − 1) = 3 × 255 = 765. For 0.5, 0.25, 0.125... (a = 0.5, r = 0.5): S∞ = 0.5/(1 − 0.5) = 1.

What's the key thing to understand about Real-World Geometric Sequences?

Compound interest: principal P at rate r% per year for n years → amount = P(1 + r/100)ⁿ. This is a geometric sequence with a = P and common ratio = (1 + r/100). Radioactive decay: each half-life, activity halves → a geometric sequence with r = 0.5. Population growth (geometric): a population growing at a constant percentage rate forms a geometric sequence. Bacterial doubling: starting with 100 bacteria doubling every 30 minutes for 5 hours (10 doublings): 100 × 2¹⁰ = 102,400. The geometric sequence model explains why bacterial populations, compound interest, and viral spread can all grow so dramatically faster than intuition initially suggests.

What do I need to know about Sum to Infinity?

The sum to infinity exists only when |r| < 1 (the series converges — terms get smaller and smaller). S∞ = a/(1 − r). For r ≥ 1: the series diverges (grows without bound). For r ≤ −1: the series oscillates and does not converge. Example: 1 + 1/2 + 1/4 + 1/8 + ... (a = 1, r = 1/2): S∞ = 1/(1 − 1/2) = 2. This means if you walked halfway across a room, then half the remaining distance, then half again... you would cover exactly 2 metres (this is Zeno's paradox resolved by geometric series).

What's the difference between Geometric and Arithmetic Sequences?

Arithmetic sequence: adds a constant (d) each time. aₙ = a + (n−1)d. Sum = n/2 × (2a + (n−1)d). Geometric sequence: multiplies by a constant (r) each time. aₙ = a × r^(n−1). Test: divide consecutive terms → constant = geometric. Subtract consecutive terms → constant = arithmetic. Interest: simple interest is arithmetic; compound interest is geometric — explaining why compound interest grows so much faster over long periods. Identifying which type: from 2, 6, 18, 54: ratios are 3, 3, 3 → geometric with common ratio 3; from 2, 6, 10, 14: differences are 4, 4, 4 → arithmetic with common difference 4.

What should I know about a Worked Example: Finding a Term and a Sum?

Take the geometric sequence 3, 6, 12, 24, ... with first term a = 3 and common ratio r = 2. To find the 10th term, use aₙ = a·r^(n-1): a₁₀ = 3 × 2⁹ = 3 × 512 = 1,536. To find the sum of the first 10 terms, use Sₙ = a(rⁿ - 1)/(r - 1): S₁₀ = 3(2¹⁰ - 1)/(2 - 1) = 3 × 1023 = 3,069. Notice how quickly the sum grows compared to the terms themselves — this is the hallmark of exponential growth. If instead r were a fraction, say r = 1/2 starting from a = 100, the terms would shrink toward zero (100, 50, 25, 12.5, ...) while the sum would approach a finite limit, which is exactly the idea behind the sum to infinity for |r| < 1.

What's the key thing to understand about a Common Ratio Less Than Zero (Alternating Sequences)?

When the common ratio r is negative, the sequence alternates in sign while the magnitude still grows or shrinks according to |r|. For example, with a = 2 and r = -3, the sequence is 2, -6, 18, -54, 162, ... — each term flips sign but grows in size because |r| = 3 > 1. If |r| < 1 and negative, such as r = -0.5, the terms alternate sign but shrink toward zero (8, -4, 2, -1, 0.5, ...), and the sum to infinity formula still applies since it only requires |r| < 1, not r itself to be positive. Alternating geometric sequences show up in physics (damped oscillations sampled at fixed intervals) and in finance when modeling alternating gains and losses that shrink over time.

What should I know about Solving for n Using Logarithms?

The geometric sequence formula aₙ = a₁ × r^(n−1) lets you find any term directly if you know its position n, but the reverse problem — finding which term number n reaches a particular target value — requires logarithms, since n appears as an exponent. Rearranging aₙ = a₁ × r^(n−1) to isolate n gives r^(n−1) = aₙ/a₁, and taking the logarithm of both sides (any base works, as long as it is applied consistently) turns the exponent into a multiplier: (n−1)log(r) = log(aₙ/a₁), so n = 1 + log(aₙ/a₁)/log(r). This technique answers practical questions such as 'how many years until an investment growing at a fixed rate doubles' or 'how many bacterial generations until a colony reaches a target size,' both of which are geometric sequence problems in disguise. For example, finding how many terms of the sequence 3, 6, 12, 24, … (first term 3, ratio 2) are needed to exceed 1,000 means solving 3 × 2^(n−1) ≥ 1000, giving n−1 ≥ log(333.3)/log(2) ≈ 8.38, so n = 10 is the first term exceeding 1,000. Because n must be a whole number, this kind of problem typically needs rounding up to the next integer rather than to the nearest one.

What should I know about a Geometric Sequence Formula Reference?

What should I know about Repeating Decimals as Geometric Series?

Sum to infinity has a neat application in converting a repeating decimal into an exact fraction. Take 0.7272727... (the digits '72' repeating forever). This is the geometric series 0.72+0.0072+0.000072+..., with first term a=0.72 and common ratio r=0.01. Since |r|<1, S∞ = a/(1−r) = 0.72/0.99 = 72/99 = 8/11. Checking: 8÷11 = 0.727272..., confirming the conversion. This is the standard method for converting any repeating decimal into an exact fraction, and it's also the proof behind the fact that 0.999... = 1 exactly: writing 0.999... as 0.9+0.09+0.009+... gives a=0.9, r=0.1, so S∞ = 0.9/0.9 = 1.

What should I know about Finding the Common Ratio from Non-Consecutive Terms?

Sometimes a problem gives two non-adjacent terms of a geometric sequence rather than two consecutive ones. If the 3rd term is 20 and the 6th term is 160, note that a₆ = a₃ × r³, since moving from the 3rd to the 6th term multiplies by r three times. So 160=20×r³, giving r³=8, so r=2. Once r is known, a₃=a×r² → 20=a×4 → a=5. This generalises: given terms a_m and a_n (n>m), r = (a_n/a_m)^(1/(n−m)) — the (n−m)th root of the ratio between the two given terms.

What should I know about the Geometric Mean of Two Numbers?

The geometric mean of two positive numbers a and b is √(ab) — the single value that, inserted between them, makes all three numbers (a, geometric mean, b) form a three-term geometric sequence. For example, the geometric mean of 4 and 16 is √(4×16) = √64 = 8, and indeed 4, 8, 16 is a geometric sequence with common ratio 2. This differs from the more familiar arithmetic mean (a+b)/2, which for the same numbers gives (4+16)/2=10 — the geometric mean (8) is always less than or equal to the arithmetic mean (10) for positive numbers, a result known as the AM-GM inequality, with equality only when a=b. The geometric mean is the more appropriate average whenever quantities combine multiplicatively rather than additively — such as averaging growth rates or investment returns over multiple periods, where using the arithmetic mean of percentage returns actually overstates the true average growth rate, while the geometric mean of the growth factors gives the accurate figure. This is why financial analysts specifically compute the geometric mean annual return (sometimes called the compound annual growth rate, or CAGR) rather than a simple arithmetic average when summarising multi-year investment performance.