high school Arithmetic Sequence

Arithmetic Sequence

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. Change the first term and the common difference and the sequence and its sum shift.

Arithmetic Sequence — Equal-step Stairs

First term a₁2

Common difference d3

Number of terms n5

💡 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

a_n = 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

Arithmetic Sum

S_n = n(a₁ + a_n)2

(first term + last term) × n / 2

In terms of d

S_n = n2(2a₁ + (n-1)d)

Sum from a₁ and d without a_n

💡 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

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

④From the next term on, recover a_n = S_n - S_{n-1}

Wrap-up

Arithmetic Core

a_n = a₁ + (n-1)d, S_n = n(a₁ + a_n)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.

Substitute into the general term a_n = a₁ + (n−1)d.

a₁₀ = 3 + (10-1) × 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.

First find the 10th term: a₁₀ = 2 + 9 × 3 = 29.

a₁₀ = 2 + (10-1) × 3 = 29

Substitute into the sum formula S_n = n(a₁ + a_n)/2.

S₁₀ = 10(2 + 29)2 = 155

▸ sum = 155

The sum is (first term + last term) × number of terms ÷ 2.

exam-style

In an arithmetic sequence {a_n}, a₃ = 7 and a₇ = 19. Find the first term a₁ and 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

Set up a₃ = a₁ + 2d = 7 and a₇ = a₁ + 6d = 19.

a₁ + 2d = 7, a₁ + 6d = 19

Subtracting gives 4d = 12, so d = 3 and a₁ = 1.

4d = 12 ⟹ d = 3, a₁ = 1

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Trigonometric Equations

Geometric Sequence

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