What is a geometric series?

A geometric series adds terms that each come from multiplying the previous term by the same number, called the common ratio. In 3 + 6 + 12 + 24 the ratio is 2, and the sum is 45. With first term a, ratio r and n terms, the sum is a(rⁿ − 1)/(r − 1), so here 3 × 15/1 = 45. The geometric level here uses the simplest case: 1 + 2 + 4 + ⋯, with ratio 2.

How do you find the sum of an arithmetic series?

You find the sum of an arithmetic series from three facts: the first term, the last term and how many terms there are. If the last term is missing, build it from the common difference d, the fixed gap between terms: aₙ = a₁ + (n − 1)d. For 4 + 7 + 10 + ⋯ with 10 terms, the last term is 4 + 9 × 3 = 31, so the sum is 10 × 35, halved: 175.

How do you remember the series for eˣ?

Remember it as matching pairs: each power of x sits over the factorial of the same number. So eˣ = 1 + x + x²/2! + x³/3! + ⋯, where n!, read “n factorial”, means 1 × 2 × ⋯ × n. The x⁵ term therefore sits over 5! = 120. Put x = 1 and the terms add up to e itself, about 2.72.

Why do students get the number of terms in a series wrong?

Students often divide the distance by the step and forget to count the first term. From 5 to 50 in steps of 3 there are 15 jumps, since 50 − 5 = 45 holds 15 threes, but there are 16 terms. Think of fence posts: 15 gaps need 16 posts. So 5 + 8 + 11 + ⋯ + 50 has 16 terms and sums to 440, not 412.5.

How could a teacher start a series lesson with this game, and is it free?

A teacher could start with a race at the board, and the game is free: every level loads in a browser, with no sign-up or download. As arithmetic series practice, write 1 + 3 + 5 + 7 + ⋯ and have the class call out each running total: 1, 4, 9, 16, 25. Once they spot the squares, ask for the sum of the first 30 odd numbers: 900. On the schools plan, priced per student, a dashboard tracks who gets better at series.

Math practice · reimagined

Series and Sums Game

This series and sums challenge has you add as many as 40 evenly spaced terms at once, then climbs to the e^x series, where x⁷ sits over 5040. Term-by-term adding is too slow here: the first right total sends your fighter in.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is a series in math?

A series is the sum of a sequence: the numbers of an ordered list, called terms, added into one total. The sequence 3, 5, 7, 9 has four terms, but the series 3 + 5 + 7 + 9 is one number, 24. Series can be short, long or endless, and a formula often gives the total without adding every term.

Every level of Series

Example: 1 + 2 + 4 + ⋯ with 5 terms

Each term doubles the one before, so the five terms are 1, 2, 4, 8 and 16.
Add them one at a time: 1 + 2 = 3, then 3 + 4 = 7, 7 + 8 = 15 and 15 + 16 = 31.
Check with the shortcut for this pattern: 2⁵ = 32, and the sum is one less, 31.
The sum is 31, so you type 31 on the keypad.

Learn it step by step

A series is what you get when you add a sequence up. Adding evenly-spaced numbers one at a time is slow, and there is a much better way.

The story goes that a schoolboy named Gauss was told to add 1 to 100 and answered almost at once. He had noticed that the numbers pair up from the outside in: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101. Fifty pairs, each worth 101, giving 5050.

The general rule is the same idea: S = n × (a₁ + aₙ) ÷ 2. Read it as "how many terms, times the average of the first and the last", because in an evenly-spaced list the average really is just the midpoint of the two ends. This is an ARITHMETIC series — every term is the last one plus the same step — and here that step is often a fraction, which does not matter: the rule only needs the first term, the last term and how many there are.

Add the first and last term, multiply by the number of terms, halve it, and type the sum.

Tips and common mistakes

To add 1 + 2 + ⋯ + n, multiply n by the next number and halve it: for 1 + 2 + ⋯ + 12, that is 12 × 13 = 156, halved to 78.
A common mistake is hunting for the gap between terms. With a₁ = 15, aₙ = 44 and n = 6 the gap is 5.8, but the sum is 6 × 59, halved: 177.
Off-screen, race a friend: one adds 2 + 4 + 6 + ⋯ + 20 term by term, the other halves 10 × 22. Both should reach 110.

How to play

On level one, each round shows a sum to finish, such as Sum: a₁=13, aₙ=37, n=12, giving its first term, last term and count, so you type 300. Another level plays like a geometric series game: 1+2+4+⋯, 7 terms, sum=? wants 127. Beat your rival to the total and Tomato strikes, and a fast total turns into a heavy blow.

— Quick answers —

Series: the short FAQ

What is a geometric series?: A geometric series adds terms that each come from multiplying the previous term by the same number, called the common ratio. In 3 + 6 + 12 + 24 the ratio is 2, and the sum is 45. With first term a, ratio r and n terms, the sum is a(rⁿ − 1)/(r − 1), so here 3 × 15/1 = 45. The geometric level here uses the simplest case: 1 + 2 + 4 + ⋯, with ratio 2.
How do you find the sum of an arithmetic series?: You find the sum of an arithmetic series from three facts: the first term, the last term and how many terms there are. If the last term is missing, build it from the common difference d, the fixed gap between terms: aₙ = a₁ + (n − 1)d. For 4 + 7 + 10 + ⋯ with 10 terms, the last term is 4 + 9 × 3 = 31, so the sum is 10 × 35, halved: 175.
How do you remember the series for eˣ?: Remember it as matching pairs: each power of x sits over the factorial of the same number. So e^x = 1 + x + x²/2! + x³/3! + ⋯, where n!, read “n factorial”, means 1 × 2 × ⋯ × n. The x⁵ term therefore sits over 5! = 120. Put x = 1 and the terms add up to e itself, about 2.72.
Why do students get the number of terms in a series wrong?: Students often divide the distance by the step and forget to count the first term. From 5 to 50 in steps of 3 there are 15 jumps, since 50 − 5 = 45 holds 15 threes, but there are 16 terms. Think of fence posts: 15 gaps need 16 posts. So 5 + 8 + 11 + ⋯ + 50 has 16 terms and sums to 440, not 412.5.
How could a teacher start a series lesson with this game, and is it free?: A teacher could start with a race at the board, and the game is free: every level loads in a browser, with no sign-up or download. As arithmetic series practice, write 1 + 3 + 5 + 7 + ⋯ and have the class call out each running total: 1, 4, 9, 16, 25. Once they spot the squares, ask for the sum of the first 30 odd numbers: 900. On the schools plan, priced per student, a dashboard tracks who gets better at series.

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