---
title: "Stirling formula: u28"
subject: "Numerical Method"
chapter: 4
key: "semester_4/numerical_method:stirling-u28"
source: https://www.bctengineer.com/q/semester_4/numerical_method/stirling-u28
site: BCT Engineering
---

# Stirling formula: u28

Numerical Method · Chapter 4: Interpolation

- Asked in: 2073 Bhadra, 8 marks
- Video: [Stirling's Central Difference Interpolation Formula Explained With Example (2nd Order Polynomial)](https://www.youtube.com/watch?v=rCZIb9ue98Q) — Sujoy Krishna Das
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/stirling-u28
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-4

## Question

Using Stirling formula find u28, given; u20 = 49225, u25 = 48316, u30 = 47236, u35 = 45926, u40 = 44306

## Short answer

For x = 28, take x0 = 30 and h = 5 so p = -0.4. Stirling's central difference formula using differences around x0 gives u28 = 47691.8256, approximately 47691.83.

## Answer

x = 28 lies near the middle of the table, so Stirling's central difference formula is suitable. Take x₀ = 30, h = 5, so p = (28 − 30)/5 = -0.4.

#### Difference table

| x | y | Δy | Δ²y | Δ³y | Δ⁴y |
|---|---|---|---|---|---|
| 20 | 49225 | -909 | -171 | -59 | -21 |
| 25 | 48316 | -1080 | -230 | -80 |  |
| 30 | 47236 | -1310 | -310 |  |  |
| 35 | 45926 | -1620 |  |  |  |
| 40 | 44306 |  |  |  |  |

Central differences about x₀ = 30: Δu₋₁ = −1080, Δu₀ = −1310; Δ²u₋₁ = −230; Δ³u₋₂ = −59, Δ³u₋₁ = −80; Δ⁴u₋₂ = −21.

Stirling's formula:

u(x₀ + ph) = u₀ + p·(Δu₋₁ + Δu₀)/2 + p²/2!·Δ²u₋₁ + p(p²−1)/3!·(Δ³u₋₂ + Δ³u₋₁)/2 + p²(p²−1)/4!·Δ⁴u₋₂

| Term | Coefficient | Difference | Value |
|---|---|---|---|
| u₀ | 1 | 47236 | 47236.0000 |
| p·(Δu₋₁+Δu₀)/2 | -0.4000 | -1195.0 | 478.0000 |
| p²/2!·Δ²u₋₁ | 0.0800 | -230 | -18.4000 |
| p(p²−1)/3!·(Δ³u₋₂+Δ³u₋₁)/2 | 0.0560 | -69.5 | -3.8920 |
| p²(p²−1)/4!·Δ⁴u₋₂ | -0.0056 | -21 | 0.1176 |

u₂₈ = 47691.8256

**Answer: u₂₈ ≈ 47691.83**

## Check yourself

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

1. Why is Stirling's formula used for u28?
   A. The spacing is unequal
   B. x = 28 lies near the middle of the table
   C. x = 28 is at the end of the table
   D. It is the only formula that gives integers
   Answer: B. Stirling's central difference formula suits a point near the centre; here x0 = 30. (Key point: Choice of formula)

2. With x0 = 30 and h = 5, what is p for x = 28?
   A. -2
   B. 2
   C. 0.4
   D. -0.4
   Answer: D. p = (28 - 30)/5 = -0.4. (Key point: Value of p)

3. In Stirling's formula the first-order term uses which quantity?
   A. Only Δu0
   B. Sum of all first differences
   C. Δu(-1) minus Δu0
   D. Mean of Δu(-1) and Δu0, i.e. -1195
   Answer: D. The term is p times (Δu-1 + Δu0)/2 = -0.4 × (-1195) = 478. (Key point: Mean first difference)

4. What is the final result for u28?
   A. About 46500
   B. About 47691.83
   C. About 48316
   D. About 47236
   Answer: B. Summing all the terms gives 47691.8256. (Key point: Final result)

## Related in Numerical Method

- [Stirling formula: y(35)](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Astirling-y35) — Numerical Method, chapter 4
- [Newton forward and Stirling](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Anewton-forward-stirling) — Numerical Method, chapter 4