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:

{`{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

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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: s_n = n/2(a₁+a_n)

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

4. nth term of geometric sequence: a_n = a₁r^n-1

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

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

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

8. Sum of geometric series: s_n = a₁ - a_nr/1-r

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