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Math SATs- Sequences and Series

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Math SATs- Sequences and Series

Welcome to a Math SAT Preparation Lesson. In this lesson we will study sequences and series.

Let's start off with sequences

A sequence is a list of numbers that has a certain order. For example, 16,7,888,-2,0,72 is a sequence, since 16 is the first number in the list, 7 is the second, etc. Sequences can have a certain "rule" by which terms progress, but they can also be completely random.

A few examples of cool sequences I can think of off the top of my head:

1,4,9,16… a sequence of the perfect squares starting from 1.

1,1,2,3,5,8,13,21,34,... the Fibonacci sequence, where each term is the sum of the previous two terms.

1, \frac{1}{2}, \frac{1}{3},... the harmonic sequence.

0,0,0,0… the "empty" sequence.

There are 2 special types of sequences we like to look at- arithmetic ones and geometric ones.

Arithmetic sequences are sequences that start with any number a, and in which every n^ {th} term can be written as a_n=a+(n-1)d, where d is any number. An example of such a sequence would be 5, 12, 19, 26, 33…, where a=5 and d=7. This is an increasing arithmetic sequence, as the terms are increasing. Decreasing arithmetic sequences have d<0.

Geometric sequences also start with any number a (though usually a is nonzero here), but this time we are not adding an extra d value each time- we multiply a by a factor of r. Thus, the n ^ {th} term is a_n=ar ^ {n-1}. Geometric sequences can either be monotonic, when r is positive and the terms are moving in one direction, or alternating, where r<0 and the terms alternate between positive and negative values, depending on n.

Sequences are fun, but they have an important application- series.

A series is a sequence of numbers that represent partial sums for another sequence. For example, if my sequence is 1,2,3,4… then my series would be 1,1+2,1+2+3,..., or 1,3,6,10….

With arithmetic and geometric series, we can use a formula to calculate any n^{th} term of the series. These shortcuts are useful since they save you from having to write out the entire sequence and add all of its terms up.

The formula for an arithmetic series is:

s_n=\frac{n(2a+(n-1)d)}{2}

The formula for a geometric series is:

s_n\frac{a(1-r ^ {n})}{1-r}

These formulas are worth memorizing.

The trick in solving sequence and series problems is recognizing first the type of sequence you're dealing with, and then finding the proper a, d or r if you're dealing with an arithmetic or geometric ones.

Are you a king of sequences?

Can you spot the patterns and calculate the n ^ {th} term? Check out the Sequence and Series test.

bobo1522



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  1. arjun7391 saidWed, 04 Jun 2008 03:34:04 -0000 ( Link )

    ya its kool>> i like it…. but wheres da harmonic mean in detail>> n only frm these formulas will b asked or what….....

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  2. sami saidSat, 26 Jul 2008 13:13:25 -0000 ( Link )

    haronic mean is not in detail

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  3. Geetika saidTue, 29 Jul 2008 11:54:39 -0000 ( Link )

    Hi Guys, I m searching for Daily updated questions on diferent subjects. If anyine knows where I can get those Questions, Let me know. Thanks

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  4. kausar saidTue, 05 Aug 2008 19:01:24 -0000 ( Link )

    thats a good and easy to learn part of serires, but it is beter to give some detail and multiple examples

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  5. cis117 saidThu, 11 Sep 2008 15:14:25 -0000 ( Link )

    it is easy to find the meaning of harmonic sequence, but it is difficult to find the examples…the book on our school does not contain the harmonic sequece

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  6. oLahav saidThu, 11 Sep 2008 15:36:04 -0000 ( Link )

    There is only 1 harmonic sequence, since it’s a name of a specific sequence and not a type of sequence. The sequence is defined as the nth term being equal to 1/n. So it’s basically 1, 1/2, 1/3, 1/4, etc. The terms get closer and closer to 0, but they never reach 0, as n grows to infinity.

    There are many applications to the harmonic sequence and if you go on to study math or science at a higher institution you’re likely to see more of it. However, in the SAT, it will most likely not appear at all, or else for one minor identification or pattern recognition question.

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  7. priyankar0105 saidSat, 11 Oct 2008 14:22:03 -0000 ( Link )

    Pls chk the formula of the sum of the GP…......i believe the correct formula is S = a(1-r^n)/(1-r)

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