Backward Difference Operator Relation with Forward |UPSC ISS 2024 Paper-1 | Problem-67 | RitwikMath

In this video, we solve an important **Numerical Analysis MCQ** involving the relationship between the **Backward Difference Operator** and the **Forward Difference Operator**.

### Problem Discussed

Find the value of:

\[
\nabla^5 y_n
\]

using the relationship between backward and forward differences.

### Concepts Covered

- Forward Difference Operator (\(\Delta\))
- Backward Difference Operator (\(\nabla\))
- Finite Difference Relations
- Operator Identities in Numerical Analysis
- Difference Calculus
- Numerical Analysis MCQ

### Important Identity Used

\[
\nabla y_n = \Delta y_{n-1}
\]

By repeated application:

\[
\nabla^r y_n = \Delta^r y_{n-r}
\]

For \(r = 5\):

\[
\nabla^5 y_n = \Delta^5 y_{n-5}
\]

### Key Learning

This problem helps understand how backward and forward difference operators are connected through index shifting.

### Useful For

- UPSC Statistics Optional
- ISS Exam Preparation
- CSIR NET Mathematical Sciences
- GATE Mathematics / Statistics
- IIT JAM Mathematics
- Engineering Mathematics
- Numerical Analysis Students

Watch till the end for a quick conceptual shortcut and exam-oriented explanation.

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