A page of formulas is useful only if you can tell which one fits the question. I start revision by checking how the terms change: do they have a constant difference, a constant ratio, or both? This sequence and series formulas revision guide puts that decision before the calculations.
All equations below use plain text, so you can copy and paste them directly into Google Docs. In every section, n is a positive integer, a is the first term, and Sₙ is the sum of the first n terms. A sequence lists terms; a series adds them. OpenStax gives a fuller reference for sequence and series equations.
Arithmetic Progression Formulas: Look for a Constant Difference
In an arithmetic progression (AP), subtracting each term from the next gives the same number, d. For example, 4, 7, 10, 13, … has d = 3.
Find a Term or a Finite Sum
Nth term: aₙ = a + (n − 1)d
Sum of n terms: Sₙ = n[2a + (n − 1)d]/2
Sum when the last term is known: Sₙ = n(a + l)/2
Here, l = aₙ. The second sum formula is quicker when a question gives you the first and last terms. For 4 + 7 + 10 + 13, use n = 4, a = 4, and l = 13:
S₄ = 4(4 + 13)/2 = 34
The arithmetic mean of two numbers sits halfway between them:
Arithmetic mean: AM = (a + b)/2
Geometric Progression Formulas: Look for a Constant Ratio
In a geometric progression (GP), dividing a term by the previous nonzero term gives a constant ratio, r. The sequence 3, 6, 12, 24, … has r = 2. A negative ratio makes the signs alternate.
Find a Term or a Finite Sum
Nth term: aₙ = ar⁽ⁿ⁻¹⁾
Sum of n terms, when r ≠ 1: Sₙ = a(1 − rⁿ)/(1 − r)
Sum when r = 1: Sₙ = na
One finite-sum formula works for r above or below 1, including negative ratios. For 3 + 6 + 12 + 24:
S₄ = 3(1 − 2⁴)/(1 − 2) = 45
MathWorld’s finite geometric series reference also makes a useful distinction: a finite GP has a sum even when its ratio would prevent an infinite GP from converging.
Check Before Using the Infinite Sum
Infinite GP sum: S∞ = a/(1 − r), valid only when |r| < 1
I write the condition beside the formula in my notes. It prevents an easy exam mistake: applying it to 2 + 4 + 8 + …, whose terms keep growing. An infinite GP can converge even when r is negative, provided −1 < r < 1. MIT OpenCourseWare explains the geometric series convergence condition.
For positive a and b, the geometric mean is:
Geometric mean: GM = √(ab)
Harmonic Progression and the Three Means
A sequence is a harmonic progression (HP) when the reciprocals of its terms form an AP. For instance, 1, 1/2, 1/3, 1/4, … is an HP because its reciprocals are 1, 2, 3, 4, … .
If an HP starts at a, and its reciprocals increase by d, then:
Nth HP term: aₙ = 1/[1/a + (n − 1)d]
The reciprocal of every term must exist. Notice what d measures here: it is the common difference of the reciprocal AP, not the difference between HP terms.
For two positive numbers a and b:
Harmonic mean: HM = 2ab/(a + b)
Order of means: AM ≥ GM ≥ HM
Relationship: GM² = AM × HM
Equality holds when a = b. These are useful checks if a question asks you to calculate all three means.
Do not confuse an HP with the harmonic series. The sum 1 + 1/2 + 1/3 + … does not have a finite value, despite its terms approaching zero. LibreTexts discusses this distinction in its harmonic series lesson.
Arithmetico-Geometric Progression: Spot Both Patterns
An arithmetico-geometric progression (AGP) multiplies corresponding AP and GP terms. Its general form is:
a, (a + d)r, (a + 2d)r², (a + 3d)r³, …
Nth term: tₙ = [a + (n − 1)d]r⁽ⁿ⁻¹⁾
Finite and Infinite AGP Sums
The finite formula needs careful brackets. The finite AGP expression in the supplied notes was incorrect; use this corrected version when r ≠ 1:
Finite AGP sum:
Sₙ = a(1 − rⁿ)/(1 − r) + dr[1 − nr⁽ⁿ⁻¹⁾ + (n − 1)rⁿ]/(1 − r)²
When r = 1: Sₙ = na + dn(n − 1)/2
Infinite AGP sum: S∞ = a/(1 − r) + dr/(1 − r)², valid when |r| < 1
For 1 + 2/3 + 3/9 + 4/27 + …, identify a = 1, d = 1, and r = 1/3. Since |r| < 1:
S∞ = 1/(1 − 1/3) + (1/3)/(1 − 1/3)²
S∞ = 3/2 + 3/4 = 9/4
When I forget the finite AGP formula, I write Sₙ and rSₙ on separate lines, then subtract. Their matching powers line up, leaving a geometric sum. That method also helps me catch a missing power of r.
Standard Sums Worth Memorizing
These three formulas handle many questions that do not present themselves as an AP or GP:
First n natural numbers: 1 + 2 + … + n = n(n + 1)/2
First n squares: 1² + 2² + … + n² = n(n + 1)(2n + 1)/6
First n cubes: 1³ + 2³ + … + n³ = [n(n + 1)/2]²
The cube formula completes the cut-off section of the supplied revision notes. Its right-hand side is the square of the sum of the first n natural numbers—a useful memory cue. LibreTexts lists these standard sums alongside series formulas.
Frequently Asked Questions
1. How do I tell an arithmetic sequence from a geometric one?
Check consecutive terms: a constant difference signals AP; a constant ratio signals GP.
2. Can I use a geometric sum formula when r is greater than 1?
Yes for a finite GP. The infinite-sum formula requires |r| < 1.
3. What is the easiest way to remember an AGP sum?
Remember the shift-and-subtract method: write Sₙ, write rSₙ, then subtract.
4. Does an HP always have a finite infinite sum?
No. Being an HP describes its terms; it does not guarantee that their infinite series converges.
Make the Formula Earn Its Place
My final check is always the same: identify the pattern, label a, d, r, and n, then test any convergence condition. Next, try one short example of each type without looking at the formulas. If you can explain why a formula applies, you are ready to use this sequence and series formulas revision guide under exam pressure.
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