Arithmetic n-th term

Deep Breakdown

What it tells me (Situation Awareness): An arithmetic sequence has a constant difference $d$ between consecutive terms. The formula $u_n = a + (n-1)d$ gives the $n$-th term in $O(1)$ time from $a$, $d$, and $n$. In DA/DS: np.linspace and np.arange produce arithmetic sequences; straight-line depreciation models a linear decrease; cohort analysis uses weekly/monthly arithmetic offsets; histogram bin edges are arithmetic sequences. Plotting $u_n$ vs. $n$ gives a straight line with slope $d$ — it is the discrete analogue of $f(x) = a + dx$.

Flowchart

Real-world use

Where it is used

Equally spaced time indices; straight-line depreciation; linearly spaced feature bins; cohort offsets (weeks 1, 8, 15, ...); uniform sampling grids; constant-rate growth/decline models.

Problem it solves

Jump directly to the $n$-th value in a linearly growing/declining sequence without iterating all prior terms.

How it solves the problem

Start at $a$, add $d$ exactly $(n-1)$ times: $u_n = a + (n-1)d$.

Example in a business

Straight-line depreciation: asset purchased for EUR 50,000 ($a = 50{,}000$, $d = -5{,}000$/yr). Book value in year $n$: $u_n = 55{,}000 - 5{,}000n$. At year 10: $u_{10} = 5{,}000$ (salvage value).

The formula, piece by piece

Each symbol:

• $u_n$: the $n$-th term
• $a$: the first term ($u_1$)
• $n$: position in sequence ($n \geq 1$, 1-based)
• $d$: common difference; $d = u_{k+1} - u_k$ for any $k$ Find $d$: $d = \dfrac{u_n - u_m}{n - m}$ Find $n$: $n = \dfrac{u_n - a}{d} + 1$ 0-based (Python indexing): $u[i] = a + i \cdot d$ for $i = 0, 1, 2, \ldots$

The formula in words

"Start at $a$. Take $(n-1)$ equal steps of size $d$. You land on $a + (n-1)d$."

Critical Concepts

• Verify constant $d$: np.diff(sequence) should all be equal.
• $d < 0$: decreasing sequence (depreciation, declining metrics).
• $d = 0$: constant sequence.
• Linear in $n$: $u_n$ is an affine function of $n$ with slope $d$.
• np.linspace: np.linspace(a, b, n) gives $n$ arithmetic terms, $d = (b-a)/(n-1)$.
• np.arange: np.arange(a, b, step) uses explicit $d$.

Step-by-step calculation (tiny example)

Sequence: 3, 7, 11, 15, ... Find $u_{20}$.

Step 1: Identify parameters $a = 3$, $d = 7 - 3 = 4$ (verify: $11-7=4$)

Step 2: Apply formula $u_{20} = 3 + (20-1)(4) = 3 + 76 = 79$

Step 3: Check $u_{20} - u_1 = 76 = 19 \times 4 = (20-1)d$

Code examples

Beginner example (Python)

import numpy as np

def arithmetic_nth_term(a: float, d: float, n: int | np.ndarray) -> float | np.ndarray:
    """n-th term: u_n = a + (n-1)*d."""
    return a + (n - 1) * d   # Sequence 3, 7, 11, ...
a_val, d_val = 3.0, 4.0
print("Arithmetic sequence (a=3, d=4):")
for n in [1, 2, 5, 10, 20]:
    print(f" u_{n:2d} = {arithmetic_nth_term(a_val, d_val, n):.1f}")   # Straight-line depreciation
print("\nDepreciation (a=50000, d=-5000):")
for year in [1, 3, 5, 10]:
    print(f" Year {year:2d}: EUR {arithmetic_nth_term(50_000, -5_000, year):>9,.0f}")   # np.linspace produces arithmetic sequences
x_values = np.linspace(0.0, 1.0, 11)
print(f"\nnp.linspace(0,1,11): {x_values}")
print(f"d = {x_values[1] - x_values[0]:.2f}, all equal: {np.allclose(np.diff(x_values), 0.1)}")

Beginner example (R)

arithmetic_nth_term <- function(a, d, n) a + (n - 1) * d
a_val <- 3; d_val <- 4
cat("Arithmetic sequence (a=3, d=4):\n")
for (n in c(1, 2, 5, 10, 20)) { cat(sprintf(" u_%d = %.1f\n", n, arithmetic_nth_term(a_val, d_val, n)))
}
cat("\nDepreciation (a=50000, d=-5000):\n")
for (year in c(1, 3, 5, 10)) { cat(sprintf(" Year %2d: EUR %9.0f\n", year, arithmetic_nth_term(50000, -5000, year)))
}

Professional example (Python)

"""
Arithmetic sequences in DA/DS: cohort offsets, feature bins, depreciation.
"""
import numpy as np
import pandas as pd

def arithmetic_nth_term(a: float, d: float, n: int | np.ndarray) -> float | np.ndarray:
    """u_n = a + (n-1)*d."""
    return a + (n - 1) * d   # 1. Weekly cohort follow-up offsets: weeks 1, 8, 15, 22, ... (d=7)
cohort_weeks = arithmetic_nth_term(1, 7, np.arange(1, 9)).astype(int)
print(f"Cohort follow-up weeks: {cohort_weeks}")   # 2. Age feature bins for marketing model
age_edges = np.linspace(18, 68, 6).astype(int)
print(f"Age bin edges: {age_edges}")   # 3. Depreciation register
assets = [("Server rack", 80_000, -8_000, 10), ("Office fit-out", 120_000, -10_000, 12)]
records = []
for name, a, d, life in assets:
    for yr in range(1, life + 1):
        records.append({"asset": name, "year": yr, "book_value": arithmetic_nth_term(a, d, yr)})
df = pd.DataFrame(records)
print("\nDepreciation (selected years):")
print(df[df["year"].isin([1, 3, 5, 10])].to_string(index=False))

Source: MFB p. 160

#calculus#series-and-transforms#arithmetic-and-geometric-progressions