Series: The method of differences

Title: Series: The Method of Differences

• The method of differences can be applied to find the sum of sequences whose terms follow a specific pattern.
• It is particularly useful when working with series whose terms form arithmetic or geometric progressions.
• In this method, we subtract consecutive terms of a sequence to develop a new sequence. The process is repeated until a constant sequence is obtained.
• If the difference sequence becomes constant, this indicates that the original sequence is arithmetic. If the ratio of terms becomes constant, the original sequence is geometric.
• The method of differences also applies to polynomial sequences. The degrees of the polynomial can provide clues to approaching problems. When the differences are constant, it is a linear polynomial; if the second differences are constant, it’s a quadratic; and if the third differences are constant, it’s a cubic polynomial and so on.
• The technique can also be applied to mixed series, which are neither purely arithmetic nor purely geometric. In such cases, separate the series into arithmetic and geometric parts and evaluate separately.
• Another application of this technique is to solve difference equations—equations defined by a difference rather than a derivative.
• The summation of a series can be evaluated by adding vertically down the diagonal differences in the table rather than horizontally across the rows or columns.
• Lastly, practise is key in gaining a good command of this method. Be sure to solve a variety of problems, including those with different degrees of complexity. This will help you familiarise with the method and recognise where and how you could apply it in different situations.