How to find algebraic patterns
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SSAT Upper Level Quantitative › How to find algebraic patterns
A sequence of numbers begins:
What is the one-hundredth entry in this sequence?
Explanation
Let 
In the following number sequence, what number goes in place of the circle?
Explanation
The sequence comprises the squares of the odd integers, in order:
...
The next number, which replaces the circle, is
Define
Which of the following expressions is equal to 
Explanation
Replace 
The top row in the above diagram shows a sequence of figures. Which figure in the bottom row is the next one in the sequence?
Figure (b)
Figure (a)
Figure (d)
Figure (c)
Explanation
Going from figure to figure, the arrow is rotating one quarter of a turn clockwise each time. Therefore, in the next figure, the arrow should be pointing to the right, thereby eliminating Figures (c) and (d) as choices and leaving Figures (a) and (b).
Starting with the third figure, each number is obtained by adding the numbers in the previous two figures (this is called the Fibonacci sequence), as follows:
The number inside the next arrow is
so the correct choice is Figure (b).
Examine the above figure. In the top row, the cubes of the whole numbers are written in ascending order. In each successive row, each entry is the difference of the two entries above it - five of those entries have been calculated for you.
What is the fifth entry in the third row?
Explanation
The fifth entry in the third row is the difference of the sixth and fifth entries in the second row.
The sixth entry in the second row is the difference of 216 and 125:
The fifth entry in the second row is the difference of 343 and 216:
Now subtract these two:
The above diagram shows a sequence of figures. In the fourth figure, each of the four variables, 
What value replaces 
Explanation
In each of the crosses, the lower left and lower right entries are the sum and product of the top two entries, respectively. The top two numbers of the next cross are the same as the bottom two of the current cross. Therefore, the fourth figure has as its top two entries 41 and 330 (the bottom two of the third figure). 
The above diagram shows a sequence of figures. In the fourth figure, each of the four variables, 
What value replaces 
Explanation
In each of the crosses, the lower left and lower right entries are the sum and product of the top two entries, respectively. The top two numbers of the next cross are the same as the bottom two of the current cross. Therefore, the fourth figure has as its top two entries 41 and 330 (the bottom two of the third figure). 
Define a sequence of numbers as follows:
For all integers 
Evaluate 
Explanation
Using the definition of this particular sequence we will plug in 
33 is the correct choice.
Give the sum of the infinite geometric series that begins
Explanation
The sum of an infinite geometric series with initial term 
The initial term is 
Anya, Brian, Clark, and Donna represented Central High in a math contest. The team score is the sum of the three highest scores; Anya outscored Clark, Brian outscored Donna; Donna outscored Clark. Whose scores were added to determine the team score?
Anya, Brian, Donna
Insufficient information is given to answer the question.
Anya, Clark, Donna
Anya, Brian, Clark
Brian, Donna, Clark.
Explanation
Let 
Three inequalities can be deduced from these statements:
Anya outscored Clark:
Brian outscored Donna:
Donna outscored Clark:
The first and third statements can be combined to arrive at:
so Brian and Donna both outscored Clark. Since Anya outscored Clark also, Clark finished last among the four, and the team score was the sum of Anya's, Brian's and Donna's scores.