Arithmetic & Geometric Sequence Calculator

Sequences Guide

What do I need to know about Arithmetic Sequences?

An arithmetic sequence has a constant difference between terms: 3, 7, 11, 15, 19... (d = 4). nth term: aₙ = a + (n−1)d. Sum of n terms: Sₙ = n/2 × (2a + (n−1)d) or n/2 × (first + last). Example: a=3, d=4, n=10: aₙ = 3 + 9×4 = 39. S₁₀ = 10/2 × (6 + 36) = 5 × 42 = 210. Real-world arithmetic sequences: even numbers (a=2, d=2), salary with annual increments, steps climbed on a staircase.

What should I know about Geometric Sequences?

A geometric sequence has a constant ratio between terms: 2, 6, 18, 54... (r = 3). nth term: aₙ = a × rⁿ⁻¹. Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) when r ≠ 1. Example: a=2, r=3, n=5: aₙ = 2×3⁴ = 162. Real-world geometric sequences: compound interest (a=principal, r=1+rate), bacterial growth, folding paper (thickness doubles each fold), radioactive decay.

What should I know about Infinite Geometric Series?

A geometric series converges to a finite sum when |r| < 1. Sum to infinity: S∞ = a/(1−r). Example: 1 + 0.5 + 0.25 + 0.125 + ... (a=1, r=0.5): S∞ = 1/(1−0.5) = 2. This is why Achilles catches the tortoise — infinite geometric series can have finite sums. Diverges when |r| ≥ 1: the sum grows without bound (r=2: 1+2+4+8+... → ∞) or oscillates (r=−2).

What's the key thing to understand about Sequences in Finance?

Compound interest is a geometric sequence: a deposit of £1,000 at 5% annual interest after n years = £1,000 × 1.05ⁿ. Mortgage repayments and annuities use the sum of a geometric series to calculate total payments. Regular savings (investing the same amount each month) form an arithmetic-then-geometric structure that underpins all retirement planning calculations.

What's the difference between Arithmetic and Geometric Sequences?

The defining feature of an arithmetic sequence is a constant common difference: each term is found by adding the same fixed amount, d, to the previous term (e.g. 3, 7, 11, 15… has d = 4). A geometric sequence instead has a constant common ratio, r: each term is found by multiplying the previous term by the same fixed factor (e.g. 3, 6, 12, 24… has r = 2). This distinction changes how the sequences grow: arithmetic sequences grow linearly (the graph of term value against position is a straight line), while geometric sequences grow exponentially (the graph curves sharply upward or decays toward zero). Arithmetic sequences model situations with steady, additive change — like a savings account receiving the same fixed deposit each month, or a staircase with equal step heights. Geometric sequences model multiplicative, compounding change — like compound interest, population growth, or radioactive decay. Confusing the two is a common error: doubling versus adding a fixed amount produce very different long-term results, especially over many terms.

What should I know about the Sum of an Arithmetic Series?

The sum of the first n terms of an arithmetic sequence (called an arithmetic series) has a neat formula that avoids adding every term individually: Sₙ = n/2 × (a₁ + aₙ), where a₁ is the first term and aₙ is the nth (last) term — essentially, n times the average of the first and last term. Equivalently, since aₙ = a₁ + (n−1)d, the formula can be written as Sₙ = n/2 × (2a₁ + (n−1)d). As a worked example, the sum of the first 10 terms of the sequence 3, 7, 11, 15… (a₁ = 3, d = 4): the 10th term is a₁₀ = 3 + 9(4) = 39, so S₁₀ = 10/2 × (3 + 39) = 5 × 42 = 210. This formula is attributed in legend to a young Carl Friedrich Gauss, who reportedly summed 1 to 100 instantly by pairing the first and last numbers (1+100, 2+99, …), each pair totalling 101, giving 50 × 101 = 5,050 — the same insight the general formula generalises.

What should I know about Arithmetic Means Between Two Terms?

An arithmetic mean between two numbers is the term (or terms) that, when inserted between them, keeps the whole sequence arithmetic — meaning the difference between consecutive terms stays constant. To insert a single arithmetic mean between a and b, it's simply (a + b)/2, the same as the ordinary average. To insert k arithmetic means between a and b, first find the common difference needed: d = (b - a)/(k + 1), then generate each inserted term by repeatedly adding d. For example, inserting 3 arithmetic means between 2 and 18 requires d = (18-2)/4 = 4, giving the sequence 2, 6, 10, 14, 18. This idea is the arithmetic-sequence counterpart to geometric means, and it's commonly used in problems that ask you to "evenly space" a set of values between two fixed endpoints.

Worked Example: Finding a Term and a Partial Sum

Arithmetic example: with first term a1 = 7 and common difference d = 4, the 20th term is a20 = a1 + (n−1)d = 7 + 19×4 = 83. The sum of the first 20 terms is S20 = n/2 × (a1 + an) = 20/2 × (7 + 83) = 10 × 90 = 900. Geometric example: with first term a1 = 3 and common ratio r = 2, the 10th term is a10 = a1 × r⁹ = 3 × 512 = 1,536. The sum of the first 10 terms is S10 = a1(rⁿ−1)/(r−1) = 3(1024−1)/1 = 3,069 — notice how quickly a geometric sum overtakes an arithmetic one with similar starting values, because growth compounds rather than accumulates at a fixed rate.

Arithmetic vs Geometric Formulas at a Glance

Spotting Arithmetic Sequences in Everyday Situations

Arithmetic sequences show up constantly outside the classroom, and recognising the pattern quickly makes mental estimation possible. A stack of chairs where each added row costs a fixed rental fee, a car loan balance that decreases by the same amount each month on a straight-line payoff, theatre seating where each row has a fixed number more seats than the last, and a savings plan with the same fixed deposit every month, are all arithmetic sequences because they share a constant difference between consecutive terms. The quickest real-world check is: does the same number get added (or subtracted) every single step? If instead a situation grows by the same multiplier or percentage each step — a bank balance earning compound interest, or a population growing 3% a year — it is geometric, not arithmetic, even though both are 'sequences'. Mixing the two up is the single most common sequence-identification mistake.

Why the Common Difference Can Be Found From Any Two Terms

A useful shortcut when only two terms of an arithmetic sequence are known (not necessarily consecutive) is that the common difference d can be recovered from d = (aₙ − aₘ) / (n − m), where aₙ and aₘ are any two terms and n, m are their positions. For example, if the 3rd term of a sequence is 11 and the 9th term is 35, then d = (35 − 11) / (9 − 3) = 24/6 = 4. Once d is known, the first term can be found by working backwards: a₃ = a₁ + 2d, so 11 = a₁ + 8, giving a₁ = 3. This two-term shortcut avoids having to know the first term in advance, which is common in word problems that only describe two snapshots of a growing pattern (such as 'after 3 months' and 'after 9 months').

Checking Whether a List of Numbers Is Actually Arithmetic

Given a list of numbers, the quickest test for an arithmetic sequence is to subtract each term from the one after it and confirm every difference is identical. For 5, 11, 17, 23: differences are 11-5=6, 17-11=6, 23-17=6 — all equal, confirming an arithmetic sequence with common difference 6. If even one difference breaks the pattern, the sequence is not arithmetic, and it's worth checking instead whether consecutive ratios (rather than differences) are constant, which would instead indicate a geometric sequence.