Math SATs- Sequences and Series

Posted Tue, 06 May 2008 18:10:16 -0000 by Oren Lahav

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

. 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.

Image Credits: bobo1522, akirsa, dominic

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Arjun Mahant – Wed, 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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Oren Lahav – Wed, 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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sami – Sat, 26 Jul 2008 13:13:25 -0000

haronic mean is not in detail

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Geetika – Tue, 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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Oren Lahav – Tue, 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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kausar – Tue, 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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Francis Cheng – Thu, 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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Oren Lahav – Thu, 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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priyankar0105 – Sat, 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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Suresh – Thu, 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

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

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Oren Lahav – Thu, 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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