An arithmetic sequence is built by repeatedly adding a fixed number to the previous term.
Once you spot the rule, like 2, 5, 8, 11 growing by a common difference of 3, any term is just first term + (n − 1) × common difference.
And by Gauss's idea of pairing the first and last terms, the sum of many terms folds into one tidy formula.
Here you can change the first term and the common difference to see how the sequence and its sum shift.
Arithmetic Sequence — Equal-step Stairs
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Adjust first term, common difference, and count to inspect the sequence
2
3
5
💡 Arithmetic = sequence with constant gap
①Adjacent terms differ by the same amount (= d)
②Like stairs of equal height
③d > 0 increases, d < 0 decreases, d = 0 is constant
General Term
nth Term
an = a₁ + (n-1)d
nth term = first term + (n-1) × d
📐 Meaning
①Start at a₁ and add d (n-1) times
②Corresponds to natural-number points on the line y = dx + (a₁ - d)
③Same shape as a linear function with slope d
Sum — Trapezoid Area
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Visualize the sum as a trapezoid area
Arithmetic Sum
Sn = n(a₁ + an)2
(first term + last term) × n / 2
In terms of d
Sn = n2(2a₁ + (n-1)d)
Sum from a₁ and d without an
💡 Gauss's Trick
①1 + 2 + ... + 100 = ?
②Pair head and tail: (1+100) + (2+99) + ... = 101 × 50 = 5050
③That is the n(a₁ + a_n)/2 idea
Arithmetic Mean
Arithmetic Mean
b = a + c2
If a, b, c is arithmetic, b is the mean of a and c
📐 Using the Mean
①Three numbers a, b, c arithmetic ⟺ 2b = a + c
②Middle term = average of the outer two
③Equally-spaced subseries are arithmetic too
Wrap-up
Arithmetic Core
an = a₁ + (n-1)d, Sn = n(a₁ + an)2
nth term and sum in one place
🎯 Exam Points
①nth term: a_n = a₁ + (n-1)d
②Sum: S_n = n(a₁ + a_n)/2
③Arithmetic mean: 2b = a + c
④S_n is quadratic in n (d ≠ 0) or linear (d = 0)
⑤Recover a_n via a_n = S_n − S_{n-1} (n ≥ 2)
Worked Examples & Past Exam
Example 1
Find the 10th term of an arithmetic sequence with first term 3 and common difference 2.
1
Substitute into the general term an = a₁ + (n−1)d.
a10 = 3 + (10-1) × 2
2
Compute.
= 3 + 18 = 21
▸ 10th term = 21
The general term adds the common difference (n−1) times to the first term.
Example 2
Find the sum of the first 10 terms of an arithmetic sequence with first term 2 and common difference 3.
1
First find the 10th term: a10 = 2 + 9 × 3 = 29.
a10 = 2 + (10-1) × 3 = 29
2
Substitute into the sum formula Sn = n(a₁ + an)/2.
S10 = 10(2 + 29)2 = 155
▸ sum = 155
The sum is (first term + last term) × number of terms ÷ 2.
2023 CSAT Math type, adapted
In an arithmetic sequence {an}, a₃ = 7 and a₇ = 19. Find the first term a₁ and common difference d.