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Sequences

A sequence is an ordered list of numbers. The following is a sequence of odd numbers:

1, 3, 5, 7, . . .

A term of a sequence is identified by its position in the sequence. In the above sequence, 1 is the first term, 3 is the second term, etc. The ellipsis symbol (. . .) indicates that the sequence continues forever.

Example

In sequence S, the 3rd term is 4, the 2nd term is three times the 1st, and the 3rd term is four times the 2nd. What is the 1st term in sequence S?

A.  0
B.  1/3
C.  1
D.  3/2
E.  4

We know “the 3rd term of S is 4,” and that “the 3rd term is four times the 2nd.” This is equivalent to saying the 2nd term is 1/4 the 3rd term: .

Further, we know “the 2nd term is three times the 1st.” This is equivalent to saying the 1st term is 1/3 the 2nd term: .

Hence, the first term of the sequence is fully determined:

1/3, 1, 4

The answer is (B).

Example

Except for the first two numbers, every number in the sequence –1, 3, –3, . . . is the product of the two immediately preceding numbers. How many numbers of this sequence are odd?

A.  one
B.  two
C.  three
D.  four
E.  more than four

Since “every number in the sequence –1, 3, –3, . . . is the product of the two immediately preceding numbers,” the forth term of the sequence is –9 = 3(–3).

The first 6 terms of this sequence are

–1, 3, –3, –9, 27, –243, . . .

At least six numbers in this sequence are odd: –1, 3, –3, –9, 27, –243.

The answer is (E).