{"id":2192,"date":"2012-06-20T10:05:48","date_gmt":"2012-06-20T17:05:48","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=2192"},"modified":"2020-01-15T10:50:39","modified_gmt":"2020-01-15T18:50:39","slug":"sequences-on-the-gmat","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/sequences-on-the-gmat\/","title":{"rendered":"Sequences on the GMAT"},"content":{"rendered":"

Understand how to handle these tricky upper level Quant problems!\u00a0<\/strong><\/p>\n

Definitions<\/strong><\/h2>\n

A sequence is a list of numbers that follow some mathematical patterns.\u00a0 More formally, a sequence is a function whose inputs are limited to the positive integers.\u00a0 Terms are denoted by a letter for the whole sequence, and in the subscript, the index<\/strong>, which is the place on the list.\u00a0 Here are some examples of common sequences.\u00a0 First, (A) and (B) are arithmetic sequences<\/strong>:<\/p>\n

A) \"a_1=, \"a_2=, \"a_3=, \"a_4=, \"a_5=, …<\/p>\n

B)\u00a0\"t_1=, \"t_2=, \"t_3=, \"t_4=, \"t_5=, …<\/p>\n

In an arithmetic sequence, the same “common difference<\/strong>” is added or subtracted each time.\u00a0 In (A), we add 3 to each term to get the next.\u00a0 In (B), we subtract 4 to get each new term.\u00a0 If you graphed the terms of an arithmetic sequence against the indices, the dots would follow a straight line in the x-y plane.<\/p>\n

The next two, sequences (C) and (D), are geometric sequences<\/strong>:<\/p>\n

C) \"e_1=, \"e_2=, \"e_3=, \"e_4=, \"e_5=, …<\/p>\n

D)\u00a0\"y_0=, \"y_1=, \"y_2=, \"y_3=, \"y_4=, …<\/p>\n

In a geometric sequence, the same “common ratio<\/strong>” is multiplied or divided each time.\u00a0 In (C), we multiply each term by 3 to get the next.\u00a0 In (D), we divide by 2 to get each new term.\u00a0 If you graphed the terms of an geometric sequence against the indices, the dots would follow an exponential function in the x-y plane.<\/p>\n

Notice also — (D) begins with a “zeroth” term.\u00a0 The first term of a sequence can have an index of either 1 or 0: both appear on the GMAT.<\/p>\n

There are other exotic sequences in mathematics.\u00a0 Here are two more<\/p>\n

E)\"j_1=, \"j_2=, \"j_3=, \"j_4=, \"j_5=, \"j_6=,\u00a0\"j_7=,\u00a0\"j_8=,\u00a0…<\/p>\n

F)\"p_0=, \"p_1=, \"p_2=, \"p_3=, \"p_4=, \"p_5=,\u00a0\"p_6=,\u00a0\"p_7=,\u00a0\u00a0…<\/p>\n

Both of these are famous in mathematics.\u00a0 The first is the Fibonacci sequence, which I will discuss further below.\u00a0 The second, known in math as the Partition Function (http:\/\/en.wikipedia.org\/wiki\/Partition_(number_theory)<\/a>)\u00a0gets into way way more difficult math that you need to know for the GMAT.<\/p>\n

 <\/p>\n

Explicit Series<\/h2>\n

A explicit series is a series in which the general rule for finding each term can be stated, either verbally or mathematically.\u00a0 Sometimes the GMAT will give you the general rule for a sequence in algebraic form:<\/p>\n

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