Sequences And Series Assignment Help

Introduction to Sequences

A Sequence is a set of things (usually numbers) that are in order. If the sequence goes on forever it is called an infinite sequence, otherwise it is a finite sequence. For example:

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{1, 2, 3, 4 ,...} is a very simple sequence (and it is an infinite sequence)

{20, 25, 30, 35, ...} is also an infinite sequence

{1, 2, 4, 8, 16, 32, ...} is an infinite sequence where every term doubles

{a, b, c, d, e} is the sequence of the first 5 letters alphabetically.

Different types of sequences:

  • Finite Sequences
  • Infinite Sequences
  • Geometric Sequences
  • Arithmetic Sequences
  • Fibonacci Sequences

Series:

Sequences And Series

Series give us one of the most common ways of getting sequences. When the terms of a sequence are added, we get a series.

The sequence 1, 4, 9, 16, 25 . . . .

Gives the series 1+4+19+25+. . . .

Sigma notation for a series:

A series can be described using the general term. For example 1 + 4 + 9 + 25 + . . . . . . . .100 Can be written ∑110n2 where 10 is the last value of 'n' and the 1 is the first value of 'n'.

Important Formulas for Sequence and Series:

1. nth term of an arithmetic sequence: an = a1 + (n-1)d

2. Sum of arithmetic series: sn = n/2(a1+an)

3. Sum of arithmetic series: sn = n/2(2a1 + (n-1)d)

4. nth term of geometric sequence: an = a1rn-1

5. nth term of geometric sequence: an = an-1r

6. Sum of geometric series: sn = a1(1-rn)/1-r

7. Sum of geometric series: sn = a1 - a1rn/1-r

8. Sum of geometric series: sn = a1 - anr/1-r

9. Sum of infinite geometric series: s = a1/1-r