ACT study guide Math: Sequences and Patterns
The sum of an infinite geometric sequence series with first term x and common ratio y < 1 is given by x/(1−y). The sum of a given infinite geometric series is 200, and the common ratio is 0.15. What is the second term of this series?
(According to the question stem, the sum of the sequence is given by x(1−y) . To solve this problem, first substitute the given values into the equation and solve for x: 200 = x(1−0.15) 200 = x0.85 170 = x The first term of the sequence, x, is 170. To find the value of the second term, multiply 170 by the common ratio, 0.15: 170 × 0.15 = )25.5
What two numbers should be placed in the blanks below so that the difference between the consecutive numbers is the same? 13, ___, ___, 34
(These four numbers will form an arithmetic sequence, a sequence in which each pair of successive terms differs by the same number. To find the difference, define d as that difference, 13 as the first term, and 34 as the fourth term. By definition, the second term is 13 + d. The fourth term, 34, can also be written as (13 + d + d) + d. Using that expression, obtain the equation 34 = 13 + d + d + d, or 34 = 13 + 3d. After subtracting 13 from both sides, divide by 3, which results in 7 = d. The difference is 7. Thus the second term is 13 + 7, or 20, and the third term is 20 + 7, or 27) 20, 27
The formula for the nth term of a sequence is a^n=1/5a^n−1+4. If the second term of the sequence is 26, what is the first term of the sequence?
(Using the formula, a^2=26=1/5a^1+4, and simplifying, 1/5a^1=22. Therefore, a1 =) 110.
The tenth term of an arithmetic sequence is 38, and the second term is 6. What is the value of the first term of this sequence?
(Arithmetic sequences are formed when each number in the sequence is the sum of some fixed number and the number before it. Suppose the fixed number being added to every term is x. Then for us to get from the second term to the tenth term, x must have been added to eight terms. Expressed as an equation, 6 + 8x = 38, so 8x = 32. This equation has a solution of x = 4. Finally, because the second term was found by adding 4 to the first term, the first term must have been 6 − 4 = ) 2
The nth term of a sequence is determined using the formula an = an−1 − r. If the fifth term of the sequence is 38 and the seventh term of the sequence is 50, what is the value of r?
(Based on the formula, to get from the fifth to the seventh term requires that r be subtracted twice. This yields the equation 38 − 2r = 50, which has a solution of) −6.
Which of the following equations represents a formula for the nth term of the sequence 14, 12, 10, 8, 6, 4, ...?
(By inspection, each term is found by subtracting 2 from the previous term.) an = an−1 − 2
What is the 101st term of the sequence 5, 4, 3, 0, 8, 5, 4, 3, 0, 8, ...?
(Every 5th term is 8. Therefore, the 100th term will be 8, since 100 is a multiple of 5, and the next term will be) 5
Which of the following numbers completes this sequence? 50, 40, 31, 23, 16, ______
(Examine the difference between each adjacent number in the series: 50 − 40 = 10 40 − 31 = 9 31 − 23 = 8 23 − 16 = 7 16 − x = 6) x = 10
The first 20 terms of a geometric sequence are less than 60. If the 1st term of the sequence is 1/2, which of the following is a possible value for the 2nd term?
(If the first 20 terms are all less than 60, the common ratio used to find terms must be quite small. If we check 2 (choice C) first, we see that to get from the 1st term to 2, we would have to multiply by 4. Thus the 20th term will be 12419, which is much larger than 60. Therefore, we can eliminate answer choices C, D, and E as being too large. A similar check eliminates answer choice B as well.) 1/4
The terms of a sequence are found by multiplying the previous term by 2 and then subtracting 1. Which of the following terms must be the largest in value?
(If the first term is negative, then each of the following terms will be more negative, and the first term will be the largest. If the first term is positive, then of those selections, the fourth term will be the largest.) It depends on the value of the first term
If the first term of a series is 4 and the third is 36, what is the value of the second?
(It is possible that the series is increasing by addition, but if so, the second term would be 20, the number equidistant from both 4 and 36. Since this is not an answer choice, you can assume the series is increasing by multiplication. Getting from the first to the third term requires that a number x be multiplied by the first term and then the second term. Therefore, 4x2 = 36 and x = 3. To get the second term, 4 is multiplied by 3, and the result is) 12.
The first and second terms of a geometric sequence are a and ab, in that order. What is the 643rd term of the sequence?
(Refer to the following chart to follow the pattern of the sequence. Term 1 2 3 4 ...n Term a ab ab2 ab3 ...abn − 1 Since the power of b is one less than the number of terms, the nth term will be abn-1. The 643rd term will then be ab643−1 = )ab642.
What is the 35th term of the sequence −1, 5, −2, 1, 5, 2, −1, 5,...?
(Since the sequence repeats every 6 numbers, the 6th, 12th, 18th, 24th, and 30th numbers in the sequence will be 2. Therefore the next 5 terms will be −1, 5, −2, 1, )5
Which of the following statements is NOT true about the arithmetic sequence 16, 11, 6, 1, . . . ?
(The best way to solve this problem is to use the process of elimination. Remember that you are looking for the answer that is NOT true! First, determine the common difference of the sequence: 16 + (−5) = 11; 11 + (−5) = 6; 6 + (−5) = 1. The common difference is −5, so eliminate answer choice D. Now, consider each of the remaining answer choices: Answer choice A: The common difference is −5, so the fifth term is 1 + (−5) = −4; eliminate answer choice A. Answer choice B: The sum of the first five terms is 16+11+6+1+−4 = 30; eliminate answer choice B. Answer choice C: If the fifth term is −4, the sixth term is −4 + (−5) = −9, and the seventh term is −9 + (−5) = −14. Answer choice C is correct because it is NOT true.) The seventh term is −12
The first term of a sequence is 4, and each subsequent term is found by multiplying the previous term by n. What is the 15th term of the sequence?
(The described sequence is geometric, and the nth term of any geometric series is a1 × rn−1, or the first term times the ratio that each term is being multiplied by. Here the first term is 4, and the ratio is n.) 4n14
The 1st term of an arithmetic sequence is m, and each subsequent term is found by adding n to the previous term. Which of the following expressions represents the value of the 99th term?
(The nth term of any arithmetic sequence is found with the formula an = a1 + d(n − 1), where d is the common difference. In this equation would be m, the variable n would be the difference, and it would be multiplied by 99 − 1, yielding) m + 98n.
What is the 400th term in the sequence 18, 21, 9, 24, 5, 18, 21, 9, 24, 5, ...?
(The sequence repeats every five terms, so for example, the 5th, 10th, 15th, 20th, and 25th terms will all be 5. Further, since 400 is a multiple of 5, the sequence will "restart" at 18 on the 401st term, and the 400th term will be) 5
The sum of the first four terms of an arithmetic sequence is 32. What is the value of the sum of the first and fourth terms of the sequence?
(The sum of the first n terms of any arithmetic sequence can be found using the formula n2(a1+an). In this case, 32=42(a1+a4), and a1 + a4 =) 16
The fifth term of an arithmetic sequence is 11x2, and the seventh term of the same sequence is 17x2. What is the third term of this sequence?
(To get from the fifth to the seventh term requires that the common difference of the arithmetic sequence be added twice. If this difference is d, then 11x2 + 2d = 17x2, and d = 3x2. Therefore, the third term is 11x2 − 3x2 − 3x2 =) 5x2
What is the 217th digit after the decimal point in the repeating decimal 0.3456⎯⎯⎯⎯⎯⎯⎯⎯?
(To solve this problem, recognize that the repeating decimal has four places (0.3456), and that the fourth place is occupied by the number 6. Therefore, every place that is a multiple of 4 will be represented by the number 6. Since 217 is not divisible by 4, you know that the 217th digit cannot be 6; eliminate answer choice E. Because 216 is a multiple of 4, the 216th digit will be 6. Therefore, the 217th digit must be 3, the next digit in the repeating decimal.) 3
Which of the following statements is NOT true about the geometric sequence 36, 18, 9, ...?
(To solve this problem, systematically evaluate each answer choice for correctness. From the sequence 36, 18, 9, ..., it is clear that each term is 1/2 of the preceding term; eliminate answer choice C. The fourth term is 9/2, or 4.5 and the fifth term is 4.5/2, or 2.25, so eliminate answer choices A and B. While the first three terms are evenly divided by 3, the fourth and fifth (and any term following) are not evenly divisible by 3,) Each consecutive term is evenly divisible by 3
The first term of a sequence is 2, and each subsequent term is found by adding −54 to the previous term. What is the first term that has a value less than zero?
The second term of the sequence is 2 − 5/4=3/4, and the third term of the sequence is 3/4 −5/4 = − 2/4 = − 1/2
The sum of the first five terms of an arithmetic sequence is 55. What is the value of the sixth term of the sequence if the first term is 3?
The sum of the first n terms of an arithmetic n sequence is found with the formula S^n=n/2(a1+an). Using the information given, 55=5/2(3+a5). This simplifies to a5=110/5−3=19. 19 − 3 = 16 is the total difference between the first and fifth terms, which can be divided by the number of terms across this difference (5 − 1 = 4) to find the common difference in the sequence. 16 ÷ 4 = 4, so the sixth term must be 19 + 4 = 23
What is the first term of the following geometric sequence? _____,30,15/2,1 7/8,...
The terms of a geometric series are found by multiplying the previous term by some fixed ratio. For us to get from 30 to 15/2, we would have to multiply 30 by 1/4. Therefore, 30 is 1/4 times the first term, and one-fourth of 120 is 30
The nth term of a sequence is found using the formula an=4a^n−1+1/2. If the third term of the sequence is 322.5, what is the value of the first term?
The third term is 322.5=4a^2+1/2. Solving this, we find the second term is 80.5 = a2. Using the formula, the second term can be written as 80.5=4a^1+1/2, which has a solution of 20 for a, the first term.
The terms of a geometric series are found by multiplying the previous term by 1/3. If the first term is a1, which of the following expressions represents the difference between the third and second terms in terms of a1?
The third term of this sequence can be represented by the formula a^3=a1(1/3)2, and the second term can be represented by a^2=a^1(1/3). Therefore, a3 − a2 = a^1(1/3)^2−a^1(1/3) = a^1(1/9−1/3) = a^1(1/9−3/9) = a^1(−2/9)
What is the next term after −1/3 in the geometric sequence 9, −3, 1, −1/3, ...?
To find the next term in the geometric sequence, recall that when looking at a geometric sequence, the nth term can be written (for some base number b) as bn. This sequence is decreasing in magnitude, thus the absolute value of b is less than 1. It also alternates signs, which indicates that b is negative. At this point, mathematical intuition might suggest trying b=−1/3, then calculating for what value of n the terms in the given sequence correspond (see table) 1/9
The expression in B shows how many dots are in a given row, but the question asks for "the total number of dots in the first n rows." That means when n = 5, the expression should total 30, not 10. One approach to solving this problem is to make a table like the one below, showing the number of rows and the cumulative number of dots. The total number of dots in rows 1 and 2 is 2(2 + 1); the total number of dots in rows 1, 2, and 3 is 3(3 + 1), and so on. You should be able to see that for the nth row, the total is the product of n and n + 1, or n(n + 1).
Which of the following describes the total number of dots in the first n rows of the triangular arrangement below?