What is the difference between a sequence and a series?

A sequence lists single terms a₁, a₂, a₃, … A series adds them up: sₙ = a₁ + … + aₙ. The values s₁, s₂, s₃, … form a sequence again, the sequence of partial sums. With a = 3 and d = 2 the seventh term is 15, the seventh partial sum 63.

How do you compute a finite arithmetic series?

With the little Gauss: sₙ = n · (a₁ + aₙ)/2, the number of terms times the mean of the first and last term. Equivalently sₙ = n · a + d · (n − 1) · n/2. For a = 3, d = 2 and n = 7 this gives 7 · 9 = 63.

Why does the series grow faster than the sequence?

Every partial sum contains all previous terms. For an arithmetic sequence with d > 0 the term grows linearly in n, the partial sum quadratically, because d · (n − 1) · n/2 contains an n². In the example the tower in year 21 is already 483 high, the step only 43.

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Staircase and Tower: Sequence and Series

About this app

Sequence versus series as a staircase and a tower: the steps aₖ = a + (k−1)d stand at the bottom, above them steps 1 … n fly up and stack into the tower sₙ = a₁ + … + aₙ. The tower tops form a sequence again (partial sums), base n·a and increase d·(n−1)n/2 can be coloured, plus the little Gauss and a geometric mode. Refutes "sₙ is the value in year n".

With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.

Sequence or series?

A sequence is a list of single values a₁, a₂, a₃, … – for example the production in every year. In an arithmetic sequence the same amount d is added every year: aₙ = a + (n − 1) · d. A series is the sequence of partial sums sₙ = a₁ + a₂ + … + aₙ, the total production up to year n. So a series is a sequence itself, a special one. With the little Gauss the finite arithmetic series is sₙ = n · a + d · (n − 1) · n/2 = n · (a₁ + aₙ)/2.

What you see

At the bottom is the staircase: step k is as high as aₖ. Above it stands a tower for every year. Choose a year and steps 1 to n fly up one by one and stack into the tower sₙ, at the same scale. Dots on the tower tops show that the tower heights form a sequence again; they grow quadratically while the steps grow evenly. On request the base a of every step is coloured: the tower then splits into n · a and d · (1 + 2 + … + (n − 1)).

Try it yourself

Tap a step or a tower and watch the tower being built. Use the sliders for the first step a, the increase d and the number of years. The Production example (a = 3, d = 2) gives a step of 15 and a tower of 63 in year 7. The 1 + … + n example shows the triangular numbers n(n + 1)/2, the geometric mode the geometric series a · (qⁿ − 1)/(q − 1). The Working panel puts every number into the formulas.

Frequently asked questions

What is the difference between a sequence and a series?: A sequence lists single terms a₁, a₂, a₃, … A series adds them up: sₙ = a₁ + … + aₙ. The values s₁, s₂, s₃, … form a sequence again, the sequence of partial sums. With a = 3 and d = 2 the seventh term is 15, the seventh partial sum 63.
How do you compute a finite arithmetic series?: With the little Gauss: sₙ = n · (a₁ + aₙ)/2, the number of terms times the mean of the first and last term. Equivalently sₙ = n · a + d · (n − 1) · n/2. For a = 3, d = 2 and n = 7 this gives 7 · 9 = 63.
Why does the series grow faster than the sequence?: Every partial sum contains all previous terms. For an arithmetic sequence with d > 0 the term grows linearly in n, the partial sum quadratically, because d · (n − 1) · n/2 contains an n². In the example the tower in year 21 is already 483 high, the step only 43.

Subject: Finance

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