Question 1

How do I find the common difference of an arithmetic sequence?

Accepted Answer

Subtract any term from the next term: d = aₙ − aₙ₋₁. Example: in 5, 11, 17, 23, … the common difference is 11 − 5 = 6. If you know two non-consecutive terms, use d = (aₘ − aₙ) / (m − n). For example, if a₃ = 10 and a₇ = 26, then d = (26 − 10) / (7 − 3) = 16/4 = 4.

Question 2

How is the arithmetic series sum formula derived?

Accepted Answer

The elegant derivation is attributed to Gauss. Write the sum forwards and backwards: S = a₁ + a₂ + … + aₙ and S = aₙ + aₙ₋₁ + … + a₁. Adding these: 2S = n(a₁ + aₙ), so S = n(a₁ + aₙ)/2. This works because each pair adds to the same constant a₁ + aₙ. Equivalently, S = n/2 × (2a₁ + (n−1)d) using the nth term formula.

Question 3

What is the difference between arithmetic sequence and arithmetic series?

Accepted Answer

A sequence is the ordered list of terms: a₁, a₂, a₃, … A series is the sum of those terms: a₁ + a₂ + a₃ + … Sₙ denotes the partial sum of the first n terms. The distinction matters in calculus and analysis: sequences converge to a limit, while series converge to a finite sum (if |r| < 1 for geometric) or diverge. Arithmetic series always diverge as n → ∞ (unless d = 0).

Question 4

How do I find which term equals a specific value?

Accepted Answer

Use the nth term formula and solve for n: aₙ = a₁ + (n−1)d → n = (aₙ − a₁)/d + 1. Example: in the sequence 4, 9, 14, 19, … which term equals 99? d = 5, a₁ = 4. n = (99 − 4)/5 + 1 = 95/5 + 1 = 19 + 1 = 20. The 20th term equals 99. If n is not a positive integer, the value is not in the sequence.

Question 5

Where do arithmetic sequences appear in real life?

Accepted Answer

Arithmetic sequences model: (1) Salary increases — annual raises of a fixed amount; (2) Depreciation — straight-line method loses a fixed amount per year; (3) Stadium seating — each row adds the same number of seats; (4) Loan amortization (simple interest) — equal principal payments; (5) Staircase steps — equal rise per step; (6) Clock positions — 12 positions at 30° intervals form an arithmetic sequence.

Question 6

How does an arithmetic sequence differ from a geometric sequence?

Accepted Answer

Arithmetic sequences add a constant (d) between terms: linear growth. Geometric sequences multiply by a constant (r) between terms: exponential growth. For example, 2, 5, 8, 11 is arithmetic (d=3); 2, 6, 18, 54 is geometric (r=3). Arithmetic sequences grow (or shrink) at a steady rate; geometric sequences grow faster and faster (if |r|>1) or converge to zero (if |r|<1).

Formula	Expression	Description
nth term	aₙ = a₁ + (n − 1)d	Any term given first term and common diff
Common difference	d = aₙ − aₙ₋₁	Constant gap between consecutive terms
Sum of n terms	Sₙ = n(a₁ + aₙ) / 2	Average of first and last, times count
Sum (alt form)	Sₙ = n(2a₁ + (n−1)d) / 2	Using only a₁ and d
Find n from value	n = (aₙ − a₁)/d + 1	Which position has value aₙ?
Arithmetic mean	AM = (a + b) / 2	Middle term between a and b

Sequence	a₁	d	a₁₀	S₁₀
Natural numbers 1,2,3,…	1	1	10	55
Even numbers 2,4,6,…	2	2	20	110
Odd numbers 1,3,5,…	1	2	19	100
Multiples of 5	5	5	50	275
Decreasing: 50,45,40,…	50	−5	5	275
Salary: $40K + $2K/yr	40000	2000	58000	490000

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