Asked to add 1 + 2 + ⋯ + 100, do you really add a hundred numbers? The first stumble in sequences is memorizing sum formulas without seeing why they take that shape. Once the shape makes sense, evaluating a sequence reduces to substituting numbers into two formulas — the general term and the sum.
This article derives and applies those formulas for arithmetic and geometric sequences through worked examples. Four symbols cover everything: a₁ is the first term, n the count of terms, d the common difference, r the common ratio.
Arithmetic sequences: a constant difference
When consecutive terms differ by a constant, the sequence is arithmetic (7, 10, 13, 16, … with d = 3). Two formulas apply:
- General term: aₙ = a₁ + (n − 1)d
- Sum: Sₙ = n(a₁ + aₙ) ÷ 2
The sum can also be written directly as Sₙ = n(2a₁ + (n − 1)d) ÷ 2 — identical once you substitute the general term, and shorter when aₙ is not needed on its own.
Example 1: the n-th term
For 7, 10, 13, … with a₁ = 7 and d = 3, the tenth term is
a₁₀ = 7 + (10 − 1) × 3 = 7 + 27 = 34
Example 2: the sum of that sequence
The sum of the first ten terms uses a₁₀ = 34:
S₁₀ = 10 × (7 + 34) ÷ 2 = 10 × 41 ÷ 2 = 205
The flow is always the same: find aₙ from the general term, then pour it into the sum formula.
Example 3: 1 through 100
The sum 1 + 2 + ⋯ + 100 is arithmetic with a₁ = 1, a₁₀₀ = 100 and n = 100:
S₁₀₀ = 100 × (1 + 100) ÷ 2 = 100 × 101 ÷ 2 = 5050
The shape of the formula comes from writing the sum forward and backward, then adding: every column totals a₁ + aₙ, and there are n of them — the famous Gauss pairing. “100 copies of 101, counted twice, so halve it” explains the ÷ 2.
One caution on n: “the sum through the 10th term” means n = 10, but a value-based cutoff (“terms at most 50”) needs the terms counted first. The n in the formula is the number of terms actually added.
Geometric sequences: a constant ratio
When consecutive terms have a constant ratio, the sequence is geometric (1, 2, 4, 8, … with r = 2):
- General term: aₙ = a₁ rⁿ⁻¹
- Sum (r ≠ 1): Sₙ = a₁(1 − rⁿ) ÷ (1 − r)
Example 4: the n-th term and the sum
For 1, 2, 4, … with a₁ = 1 and r = 2:
a₇ = 1 × 2⁶ = 64, S₇ = (1 − 2⁷) ÷ (1 − 2) = (1 − 128) ÷ (−1) = 127
Adding 1 + 2 + 4 + 8 + 16 + 32 + 64 by hand gives 127, matching exactly.
Ratios can be fractions: 3, 1.5, 0.75, 0.375, … has r = 1/2, and the first four terms sum to
S₄ = 3 × (1 − (1/2)⁴) ÷ (1 − 1/2) = 3 × (15/16) ÷ (1/2) = 5.625
which matches 3 + 1.5 + 0.75 + 0.375. Sequences with |r| < 1 creep toward zero, so their sums grow slowly.
The r = 1 caveat
When r = 1 every term equals the first (3, 3, 3, …), so the sum is simply Sₙ = a₁ × n. The formula divides by 1 − r, which is zero at r = 1 — tests love this case, and the switch is easy to forget.
The geometric formula itself comes from a subtraction trick: multiply the sum by r, subtract, and every middle term cancels, leaving (1 − r)S = a₁(1 − rⁿ). Arithmetic sums pair up; geometric sums cancel.
Three common mistakes: mixing up arithmetic and geometric (check first: constant difference or constant ratio?); misreading n (the sum through the 50th term has n = 50, but a value-based cutoff needs counting first); and dropping the ÷ 2 in the arithmetic formula, or using the geometric formula at r = 1. Writing out a₁, d (or r) and n before substituting prevents all of them.
Verify with the tool
For checking hand calculations, Tools Hub’s Sequence & Series Calculator takes the first term, difference (or ratio) and count, and returns the n-th term and the sum with the substituted formulas shown. Integer inputs are computed with BigInt, so even a geometric sum with about 31 digits (ratio 2, 100 terms) comes out exactly.
How to use it (3 steps)
Pick the sequence type
Choose the Arithmetic or Geometric tab — the input labels switch between d and r automatically.
Enter a₁, d (or r) and n
The preset button for 1 + 2 + ⋯ + 100 fills in a₁ = 1, d = 1, n = 100 at once.
Read the term and the sum
The Working panel shows the substitution into the general-term and sum formulas, so you can see exactly which step of your hand calculation differs.
The tool featured in this article
Sequence & Series Calculator
n-th terms and sums with the substituted formulas shown, exact BigInt arithmetic for large geometric results — free and browser-based.
To reproduce the examples, pick the Arithmetic tab and enter a₁ = 7, d = 3, n = 10 to read 34 and 205. Switch to Geometric with a₁ = 1, r = 2, n = 7 for 64 and 127.
Fractions appear naturally in geometric ratios — the guide to working with fractions covers their handling, and mean, median and standard deviation covers summarizing data.
Summary
- Arithmetic: aₙ = a₁ + (n−1)d and Sₙ = n(a₁ + aₙ) ÷ 2
- Geometric: aₙ = a₁rⁿ⁻¹ and Sₙ = a₁(1 − rⁿ) ÷ (1 − r), with Sₙ = a₁ × n when r = 1
- The ÷ 2 adjusts for pairing the sum forward and backward (Gauss)
- List a₁, d (or r) and n before substituting to avoid misreads
- n in the formula is the number of terms actually added, not necessarily a number quoted in the problem
FAQ
Why does the arithmetic sum formula divide by 2?
Adding the sum forward and backward gives two copies of the total, arranged as n pairs that each total a₁ + aₙ. Halving recovers the true sum — the 1 + 100 = 101 pairing idea.
Are geometric sequences with ratio 0 handled?
Normally they are excluded: with r = 0 every term after the first is zero, which breaks the “constant ratio” structure. The tool treats r = 0 as an error, just as mathematics conventionally does.
Can the tool compute an infinite geometric series?
No — it sums finitely many terms. An infinite geometric series with |r| < 1 uses the separate formula S = a₁ ÷ (1 − r), which is outside this tool’s scope.
Are very large sums still exact?
Yes for integer inputs: BigInt arithmetic introduces no rounding, so a ratio-2, 100-term sum (about 31 digits) is exact. Extreme ratios or term counts hit a size limit, which the tool reports.
References
- OpenStax, College Algebra 2e, Section 9.2 “Arithmetic Sequences” (general term): openstax.org
- OpenStax, College Algebra 2e, Section 9.3 “Geometric Sequences” (general term): openstax.org
- OpenStax, College Algebra 2e, Section 9.4 “Series and Their Notations” (arithmetic and geometric sums): openstax.org
