---
title: "Forward difference table"
subject: "Numerical Method"
chapter: 4
key: "semester_4/numerical_method:forward-difference-table-1-6"
source: https://www.bctengineer.com/q/semester_4/numerical_method/forward-difference-table-1-6
site: BCT Engineering
---

# Forward difference table

Numerical Method · Chapter 4: Interpolation

- Asked in: 2069 Bhadra, 4 marks
- Video: [Numerical Analysis - Construction of a Forward Difference Table - Chapter 1 -  BSC Final Year / 5th](https://www.youtube.com/watch?v=f4Uqvg-FF9k) — Dr. Vikas Poply - MATHS EASY
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/forward-difference-table-1-6
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-4

## Question

Generate forward difference table from the following data.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| f(x) | 2 | 9 | 28 | 65 | 126 | 217 |

## Short answer

The forward difference table first row is 2, 7, 12, 6, 0, 0. The third differences are constant 6 and fourth differences are zero, indicating f(x) is a cubic polynomial x³+1.

## Answer

Here h = 1. A forward difference table lists Δyᵢ = yᵢ₊₁ − yᵢ, then Δ²yᵢ = Δyᵢ₊₁ − Δyᵢ, and so on, until the differences vanish.

| x | y | Δy | Δ²y | Δ³y | Δ⁴y | Δ⁵y |
|---|---|---|---|---|---|---|
| 1 | 2 | 7 | 12 | 6 | 0 | 0 |
| 2 | 9 | 19 | 18 | 6 | 0 |  |
| 3 | 28 | 37 | 24 | 6 |  |  |
| 4 | 65 | 61 | 30 |  |  |  |
| 5 | 126 | 91 |  |  |  |  |
| 6 | 217 |  |  |  |  |  |

Working:

```text
Δf: 9−2 = 7,  28−9 = 19,  65−28 = 37,  126−65 = 61,  217−126 = 91
Δ²f: 19−7 = 12,  37−19 = 18,  61−37 = 24,  91−61 = 30
Δ³f: 18−12 = 6,  24−18 = 6,  30−24 = 6
Δ⁴f: 0, 0, ...
```

The third differences are constant (= 6) and the fourth differences are zero, so f(x) is a cubic polynomial (indeed f(x) = x³ + 1). The first row (2, 7, 12, 6, 0, 0) is the set of values used by Newton's forward formula.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How are second differences formed?
   A. y(i+1) - y(i-1)
   B. Sum of Δy
   C. Δ²yi = Δyi²
   D. Δ²yi = Δy(i+1) - Δyi
   Answer: D. Differences of differences. (Key point: Definition)

2. Third differences are constant at 6. What does that imply?
   A. f is linear
   B. f is quadratic
   C. f is exponential
   D. f(x) is a cubic polynomial
   Answer: D. Constant nth differences mean degree n. (Key point: Constant difference)

3. Which values does Newton's forward formula use?
   A. First column only
   B. Middle row
   C. Last diagonal
   D. First row: 2, 7, 12, 6, 0, 0
   Answer: D. The top diagonal gives y0, Δy0, Δ²y0, ... (Key point: First row)

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