Method of Differences
Definition:
The method of differences is a technique used in mathematics to study and analyze sequences or series by examining the differences between consecutive terms.
Basic Concept:

First Differences:
 The first differences of a sequence are obtained by subtracting each term from its consecutive term.
 If the sequence is ${a}_{1},{a}_{2},{a}_{3},\dots $, then the first differences are ${a}_{2}{a}_{1},{a}_{3}{a}_{2},{a}_{4}{a}_{3},\dots $

Second Differences:
 Similarly, the second differences are obtained by taking the differences between consecutive first differences.
 If the first differences are ${d}_{1},{d}_{2},{d}_{3},\dots $ then the second differences are ${d}_{2}{d}_{1},{d}_{3}{d}_{2},{d}_{4}{d}_{3},\dots $
Use of Method of Differences:

Identifying Sequences:
 Helps in determining whether a sequence follows an arithmetic, geometric, or other patterns.
 For example, constant first differences indicate an arithmetic sequence, constant second differences indicate a quadratic sequence, etc.

Finding Patterns:
 Useful in recognizing patterns or establishing relationships between terms in a sequence or series.

Predicting Terms:
 Can aid in predicting future terms of a sequence or series based on observed differences.
Example:
Consider the sequence: $3,6,12,24,48,\dots $

First Differences:
 $3,6,12,24,\dots $ (First differences are $3,6,12,24,\dots $)

Second Differences:
 $3,6,12,\dots $ (Second differences are $3,6,12,\dots $)
This indicates that the sequence of numbers $3,6,12,24,\dots $ forms a geometric sequence, where each term is double the previous term.
Application:
 Pattern Recognition:
 Helps in recognizing and analyzing patterns in sequences or series.
 Prediction:
 Enables prediction of future terms in a sequence based on observed differences.
 Mathematical Analysis:
 Useful in solving problems related to sequences and series in various fields of mathematics and science.