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8.1 Sequences and Summation NotationDEFINITION OF A SEQUENCE A sequence is a function f whose domain is the set of natural numbers. The terms of the sequence are the function values                              f (1), f (2),  f (3),...,  f (n),...We usually write an instead of the function notation f (n). So the terms of the sequence are written as a1, a2, a3,..., an,...The number a1 is called the first term, a2 is called the second term, an is called the nth term

A sequence is a list of numbers that have some type of pattern.A series is when we add up a sequence of numbers.

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f(x) = 2x an = 2n

The difference between a function, f(x), and a sequence, an, is the domain. For a function above the domain is all real numbers, but for the corresponding sequence, the domain is just positive integers, so we only have points. For the sequence above: a1 = 2, a2 = 4, a3 = 6, a4 = 8,...

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5, 10, 15, 20, 25,...

a1                 a2                   a3               a4                     a5,...

Here is a sequence of numbers and the corresponding terms

Explicit formula: Recursive Formula:

means "the previous term"

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Given the following sequence give the recursive and explicit formula.

2, 4, 6 ,8 ,10,...

Explicit nth term formula: Recursive formula:

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Find the first five terms and the 100th term of the given sequences:

nth term First Five Terms 100th termAre these recursive or explicit formulas?

For the first 5 terms, just plug 1, 2, 3, 4, & 5 into the an formula.

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Find the nth term of a sequence whose first several terms are given:

Look for patterns, like the denominator is increasing by 2 each time, and the numerator is 1 less than the denominator.

2, 4, 6, 8,...

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Given

Find the first five terms of the sequence

Is this a recursive or explicit formula?

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Find the 11th term of the Fibonacci SequenceIs this a recursive or explicit formula?

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S1 is called the first partial sum, S2 is the second partial sum, and so on. Sn is called the nth partial sum. The sequence S1, S2, S3,..., Sn,... is called the sequence of partial sums.

THE PARTIAL SUMS OF A SEQUENCE

For the sequence

The partial sums are a1, a2, a3,..., an,...

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Find the first four partial sums and the nth partial sum of the sequence given by

First you have to find the first four terms (a1, a2, a3, a4).

Now we can find the first four partial sums.

If we look at the partial sums, 1/2, 3/4, 7/8, & 15/16, we can see a pattern. From that pattern we can write the nth partial sum (Sn):

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Find the first four partial sums and the nth partial sum of the sequence given by

First you have to find the first four terms (a1, a2, a3, a4).

Now we can find the first four partial sums.

If we look at the partial sums, 1/2, 2/3, 3/4, & 4/5, we can see a pattern. From that pattern we can write the nth partial sum (Sn):

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SIGMA NOTATION (or Summation Notation):

The left side of this expression is read, “The sum of ak from k = 1 to k = n.” The letter k is called the index of summation, or the summation variable, and the idea is to replace k in the expression after the sigma by the integers 1, 2, 3, . . . , n, and add the resulting expressions, arriving at the right­hand side of the equation.

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Find each sum:

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When the instruction says, "Write the sum without using sigma notation." it means to just write the terms of the sum, but don't actually do the adding.

Example:

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Write each sum using sigma notation:

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PROPERTIES OF SUMSLet a1, a2, a3, a4,... and b1, b2, b3, b4,... be sequences. Then for every positive integer n any real number c, the following properties hold.

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Assignment:

Section 8.1 on Webassign