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Math SATs- Sequences and Series

by Oren Lahav

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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.

%{font-size:16px; color:purple}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.

%{color:purple}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.

Image Credits: bobo1522, akirsa, dominic

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13 Comments
    tony211
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    tony211Wed, 18 Sep 2013 10:28:04 -0000

    A geometric series has first term 1 and the common ratio r is positive. The sum of the first 5 terms of the geometric series is twice the sum of the terms from 6th to the 15th inclusive. Show that r*2 =1/2(?3 - 1).
    help me please..this question makes me crazy

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    rizvirgo
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    rizvirgoMon, 12 Nov 2012 18:55:24 -0000

    help me please to solve this series
    -1, 2, 12, 40, ? ? ? ?

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    orthohin08
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    orthohin08Mon, 01 Oct 2012 04:04:29 -0000

    N1) 2,6,3,9,6,_
    N2) 1,4,2,8,6,_

    Plz answer the two series with explanation.

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    karan makhija
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    karan makhijaThu, 02 May 2013 19:42:06 -0000
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    indian bheja
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    Tue, 02 Aug 2011 16:49:46 -0000

    make maths interesting and pleasant to see viewers .

    pls include sum topic of integration
    and clear our funda.

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    BIRUK GEBRE
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    BIRUK GEBREWed, 13 Apr 2011 08:18:40 -0000

    That a good and easy to learn

    THANKS VERY M------------------H

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    itsmevvk
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    itsmevvkWed, 16 Jun 2010 13:52:42 -0000

    please anyone tell me the sum upto 2009 terms of series :
    1/3! + 5/4! + 11/6! + 19/7! ……………..
    send the answer at my e-mail id plzzzzzz……..

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    priyankar0105
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    priyankar0105Sat, 11 Oct 2008 14:22:03 -0000

    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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    Sureshbala
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    SureshThu, 27 Nov 2008 12:08:53 -0000

    Yes, what you said is absolutely true.

    The formula to calculate the Sum to n terms in a G.P is

     S_n = a (\frac{1-r^n}{1-r}); \quad if \quad r <1

    and it is

     S_n = a (\frac{r^{n}-1}{r-1}); \quad if \quad r>1
    oLahav
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    Oren LahavThu, 27 Nov 2008 15:37:38 -0000

    Thanks for correcting me. I personally don't memorize these formulas, since they're pretty easy to derive in you have to use them. Luckily, they're not used too often in the SAT.

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    oLahav
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    Oren LahavThu, 11 Sep 2008 15:36:04 -0000

    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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    cis117
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    Francis ChengThu, 11 Sep 2008 15:14:25 -0000

    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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    kausar
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    kausarTue, 05 Aug 2008 19:01:24 -0000

    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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    Geetika
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    GeetikaTue, 29 Jul 2008 11:54:39 -0000

    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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    oLahav
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    Oren LahavTue, 29 Jul 2008 13:48:42 -0000

    You can find a lot of questions in our Question Bank, which has hundreds of original SAT questions on everything from geometry to sentence corrections.

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    sami
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    samiSat, 26 Jul 2008 13:13:25 -0000

    haronic mean is not in detail

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    arjun7391
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    Arjun MahantWed, 04 Jun 2008 03:34:04 -0000

    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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    oLahav
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    Oren LahavWed, 04 Jun 2008 13:27:54 -0000

    The harmonic sequence just means that for each n from 1 to infinity, the nth term of the sequence is 1/n.

    On the SATs you may be asked about any sort of sequence- arithmatic, geometric, even some random ones you'll have to spot yourself. The formulas given here are most frequently asked about and they'll also help you in your later studies of sequences and series in terms of calculus limits, if you go on to study math at college, so I thought including them is a good idea.

    For any other questions, post a discussion. And make sure to check out the Sequences test to know exactly what type of questions you'll be asked.

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Last Updated At Jul 04, 2013
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