An arithmetic sequence is a row of numbers made by adding the same amount each time.
2, 5, 8, 11 grows by 3, so its common difference is 3, and since the nth term is the first term plus (n − 1) common differences, you can jump straight to any term.
The sum comes from pairing: the first and last terms, the second and second-to-last, and so on all give the same total.
Change the first term and the common difference, and the staircase in the first picture and the rectangle in the second change together.
Arithmetic sequences: a staircase with equal steps
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Each bar is a term; the number at the top is the sum of the first n terms
2
3
5
💡 Neighboring terms always differ by the same amount
①Subtract any term from the next one and you always get the common difference d
②So the tops of the bars line up on one straight line
③If d > 0 the terms grow, if d < 0 they shrink, and if d = 0 every term is the same
The general term: how many times d is added
General term
an = a₁ + (n-1)d
nth term = first term + (n − 1) × common difference
📐 Reading the general term
①Start at a₁ and add d exactly (n − 1) times
②The points (n, aₙ) lie on the line y = dx + (a₁ − d), whose slope is d
③So an arithmetic sequence is a linear function that only takes whole-number inputs
The sum: flip a copy to make a rectangle
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A reversed copy stacked on the gold sequence makes a rectangle; the sum is half of it
Sum of an arithmetic sequence
Sn = n(a₁ + an)2
(first term + last term) × number of terms ÷ 2
The sum written with d
Sn = n2(2a₁ + (n-1)d)
When the last term aₙ is unknown, the first term and the common difference are enough
💡 Gauss's pairing
①In 1 + 2 + … + 100, pair 1 with 100 and 2 with 99, and every pair adds up to 101
②There are 50 pairs, so the total is 101 × 50 = 5050
③Write the sequence again in reverse and add column by column: all n columns become a₁ + aₙ, and the original sum is half of that
Arithmetic mean
Arithmetic mean
b = a + c2
If a, b, c form an arithmetic sequence in that order, b is the average of a and c
📐 Using the arithmetic mean
①a, b, c form an arithmetic sequence in that order ⟺ 2b = a + c
②The middle term is the average of its two neighbors
③Terms picked at equal gaps from an arithmetic sequence form another arithmetic sequence
④If you know the sum Sₙ, the general term comes back as aₙ = Sₙ − Sₙ₋₁ for n ≥ 2
Summary
Key formulas
an = a₁ + (n-1)d, Sn = n(a₁ + an)2
The general term and the sum side by side
🎯 Exam Points
①General term aₙ = a₁ + (n − 1)d
②Sum Sₙ = n(a₁ + aₙ)/2 = n{2a₁ + (n − 1)d}/2
③Arithmetic mean 2b = a + c
④When d ≠ 0, Sₙ is a quadratic in n with no constant term
⑤aₙ = Sₙ − Sₙ₋₁ (n ≥ 2), and a₁ = S₁
Examples and past exam
Example 1
Find the 10th term of the arithmetic sequence with first term 3 and common difference 2.
1
Put n = 10 into the general term aₙ = a₁ + (n − 1)d.
a10 = 3 + (10 − 1) × 2
2
From the 1st term to the 10th, 2 is added nine times, an increase of 18.
= 3 + 18 = 21
▸ a₁₀ = 21
Reaching the 10th term takes nine additions of the common difference, not ten, because the first term is already standing at the front of the line.
Example 2
Find the sum of the first 10 terms of the arithmetic sequence with first term 2 and common difference 3.
1
The sum formula needs the last term, so find the 10th term first.
a10 = 2 + (10 − 1) × 3 = 29
2
Multiply the first-plus-last total, 31, by the number of terms, 10, and divide by 2.
S10 = 10(2 + 29)2 = 155
▸ S₁₀ = 155
Pair 2 with 29, 5 with 26, 8 with 23, and every pair adds up to 31. Ten terms make five pairs, so the sum is 31 × 5 = 155.
2023 CSAT Math type, adapted
In an arithmetic sequence, a₃ = 7 and a₇ = 19. What are the first term a₁ and the common difference d?
①a₁ = 1, d = 3
②a₁ = 3, d = 2
③a₁ = 1, d = 4
④a₁ = 2, d = 3
⑤a₁ = 7, d = 3
▸ ① a₁ = 1, d = 3
1
The 3rd term is the first term plus the common difference twice; the 7th term adds it six times.
a₁ + 2d = 7, a₁ + 6d = 19
2
The two terms are four steps apart and differ by 19 − 7 = 12, so d = 3 and a₁ = 7 − 2 × 3 = 1.