Mathematical Algorithms - Sequence & Series

Mathematical algorithms are step-by-step procedures used to solve math problems. This article looks at sequences and series, which are important parts of these algorithms. Sequences are ordered sets of numbers, while series are the sums of these numbers. Understanding sequences and series is vital for solving complex math problems, modeling real-world situations, and developing advanced computer techniques. This article will give you a clear overview of the key ideas, uses, with some practice problems of mathematical algorithms involving sequences and series..

What is Sequence?

A sequence is an arrangement of a set of numbers in a particular order defined by some rule. If a ₁ , a ₂ , a ₃ . . . is a sequence then 1, 2, 3 denotes the position of the elements in the sequence. A sequence can be finite or infinite.

Some well-known sequences are:

• Arithmetic Sequence
• Geometric Seqeunce
• Harmonic Sequence
• Fibonacci Sequence
• Juggler Sequence
• Padovan Sequence
• Aliquot Sequence
• Moser-de Bruijn Sequence
• Stern-Brocot Sequence
• Newman-Conway Sequence
• Sylvester’s sequence
• Recaman’s sequence

What is a Series?

A series is formed by adding the elements of the sequence. If a ₁ , a ₂ , a ₃ . . . is a sequence then the series is given as a ₁ + a ₂ + a ₃ + . . . Note that the series refers to the indicated sum of the sequence, not the sum itself.

Difference between Sequence and Series:

Easy Problems on Sequence and Series:

• Tetrahedral Numbers
• Rectangular (or Pronic) Numbers
• Program to check if N is a Pentagonal Number
• Hexagonal Number
• Octagonal number
• Nonagonal number
• Decagonal Numbers
• Program to print Arithmetic Progression series
• Program for N-th term of Arithmetic Progression series
• Program to print GP (Geometric Progression)
• Program for N-th term of Geometric Progression series
• Program to find sum of first n natural numbers
• Sum of first n even numbers
• Lucas Numbers
• Tribonacci Numbers
• Keith Number
• Disarium Number
• Pell Number
• Hailstone Numbers
• Program for Perrin numbers

Medium Problems on Sequence and Series:

• Program for sum of arithmetic series
• Find N Arithmetic Means between A and B
• Program for sum of geometric series
• Find N Geometric Means between A and B
• Program for harmonic mean of numbers
• Harmonic progression Sum
• Sum of squares of first n natural numbers
• Sum of square of first n odd numbers
• Sum of square of first n even numbers
• Nicomachu’s Theorem
• Sum of cubes of first n odd natural numbers
• Sum of cubes of first n even numbers
• Squared triangular number (Sum of cubes)
• Square pyramidal number (Sum of Squares)
• Centered hexagonal number
• Program for centered nonagonal number
• Carol Number
• Woodall Number
• Taxicab Numbers
• Ludic Numbers
• Sum of the series 5+55+555+.. up to n terms

Hard Problems on Sequence and Series:

• Program for sum of cos(x) series
• Program to print binomial expansion series
• Find m-th summation of first n natural numbers
• Sum of series 2/3 – 4/5 + 6/7 – 8/9 + ——- upto n terms
• Sum of the series 0.6, 0.06, 0.006, 0.0006, …to n terms
• Sum of the series 1, 3, 6, 10… (Triangular Numbers)
• Sum of the Series 1 + x/1 + x^2/2 + x^3/3 + .. + x^n/n
• Sum of the Series 1/(1*2) + 1/(2*3) + 1/(3*4) + 1/(4*5) + . . . . .
• Sum of the series 1 + (1+2) + (1+2+3) + (1+2+3+4) + …… + (1+2+3+4+…+n)
• Sum of series M/1 + (M+P)/2 + (M+2*P)/4 + (M+3*P)/8……up to infinite
• Sum of the series 1.2.3 + 2.3.4 + … + n(n+1)(n+2)
• Sum of Series (n^2-1^2) + 2(n^2-2^2) +….n(n^2-n^2)
• Program to find the sum of a Series 1 + 1/2^2 + 1/3^3 + …..+ 1/n^n
• Program to get the Sum of series: 1 – x^2/2! + x^4/4! -…. upto nth term
• Efficient Program to Compute Sum of Series 1/1! + 1/2! + 1/3! + 1/4! + .. + 1/n!
• Program to find Sum of a Series a^1/1! + a^2/2! + a^3/3! + a^4/4! +…….+ a^n/n!
• Find n-th term of series 1, 3, 6, 10, 15, 21…
• Finding nth term of any Polynomial Sequence

Quick Links:

• Recent Articles on Series!
• Practice problems on Mathematical Algorithms