Sequences & Series Questions and Answers

Given the recursive formula shown, what are the first 4 terms of the sequence? f(n) = f(1) = 8 [f(n)=f(n-1)+7 if n>1 OA) 7, 13, 19, 25 B) 6, -1, -8, -15 C) 6, 42, 294, 2,058 D) 6, 13, 20, 27

Billy writes out all of the numbers from 1 to 103 on the board, in order from smallest to largest from left to right. He then goes down the board, starting from 1, up to 100, erasing each number and replacing it with the sum of the next two numbers on the right. For example, he replaces 1 with 2 + 3 = 5, 2 with 7, 3 with 9, and 4 with 11. He then goes back to the beginning, starting with 5, then replaces each number with the sum of the next three numbers on the right, until this is not possible anymore. For example, the 5 he wrote on the very left gets replaced with 7+9+ 11 = 27. What is the sum of the first 100 numbers on the board?

Two students collect baseball cards. Shushu has 210 cards, and Vivak has 160 cards. Shushu adds 8 new cards to her collection each month, and Vivak adds 13 new cards each month. After how many months will they have the same number of baseball cards?

Simone, Sergio, Maria, Kim, Dawn, Tyrone, and Larry have all been invited to a dinner party. They arrive randomly and each person arrives at a different time. a. In how many ways can they arrive? b. In how many ways can Simone arrive first and Larry last? c. Find the probability that Simone will arrive first and Larry last.

Jake has started an exercise program. Each day he increases the number of sit-ups that he does by a constant amount. On day one, he did 7 sit-ups. In the following three days he did 9, 11, and 13 sit-ups, respectively. On what day will he do 81 sit-ups? O 37 40 36 38 39

The observation deck of the Space Needle in Seattle, Washington, is 526 feet above the ground. A six-foot-tall man is watching a car on the street below. Let d represent the distance from the man to the car and the angle of depression. Write d as a function of 0.

What is the explicit rule for the nth term of the geometric sequence? 4, 12, 36, 108, 324, ... an = 4(3n) an = 3(4n-1) an = 4(3n-1) an = 4(3n+1)

My water bill includes a fixed cost per month plus a fee for each gallon used in a month. In February, I used 1200 gallons and paid $53 and in March I used 1850 gallons and paid $59.50. What is the fixed cost per month?

A supermarket finds that the number of boxes of a new cereal sold increases each week. In the first week, only 20 boxes of the cereal were sold. In the next week, 53 boxes of the cereal were sold and in the third week 86 boxes of the cereal were sold. The number of boxes of cereal sold each week represents an arithmetic sequence. What is the explicit rule for the arithmetic sequence that defines the number of boxes of cereal sold in week n?

Suppose you agreed to work for 1¢ the first day, 2¢ the second day, 4¢ the third day, 8¢ the fourth day, and so on, with your wages doubling each day. Determine (a) how much you will earn on day 11 and (b) how much you will have earned altogether after your wages are paid on the day indicated.

Given two terms in an arithmetic sequence, find the first five terms: a18 = -80 and a35 = -131 -29, -32, -35, -38, -41 -29, -24, -19, -14, -9 -29, -25, -21, -17, -13 -29, -33, -37, -41, -45

Find the nth degree Taylor polynomial Tn for n = 0, 1, 2, and 3 generated by the function f(x) = sin x about the point x = π/3. To(x) = T₁(x) = T₂(x) = T3(x) =

Match an expression for the apparent nth term of the sequence. (Assume n begins with 1) 1. 3, 7, 11, 15, 19, ... 2.1, 1/4,1/9,1/16,1/25,... 3. 2/1,3/3,4/5,5/7,6/9,.. 4. 1/3,-2/9,4/27,-8/81,... A. an =n+1/2n-1 B. an= (-2)^n-1/3^n C. an = 4n -1 D. an = 1/n^2

Dante is making a necklace with 18 rows of tiny beads in which the number of beads per row is given by the series 3 + 10 + 17 + 24 + a. If you were to write this series in summation notation, give i. the lower limit of the sum ii. the upper limit of the sum iii. the explicit formula of the sum b. Find the total number of beads in the necklace. Explain your method for finding the total number of beads.

You are walking along a beach and your toe hits something hard You reach down, grab onto a handle, and pull out a lamp! It is sandy. You start to brush it off with your towel Poof! A genie appears. He tells you, "Thank you for freeing me from that bottle! I was getting claustrophobic. You can choose one of these purses as a reward." • Purse A which contains $1,000 today. If you leave it alone, it will contain $1,200 tomorrow (by magic). The next day, it will have $1,400. This pattern of $200 additional dollars per day will continue. • Purse B which contains 1 penny today. Leave that penny in there, because tomorrow it will (magically) turn into 2 pennies. The next day, there will be 4 pennies. The amount in the purse will continue to double each day. 1. How much money will be in each purse after a week? After two weeks?

Let the sequence A0, A1, A2, A3,... be defined by the formula a = nteger n ≥ 0. Which of the following recurrence relation, if any, does this sequence satisfy? O This sequence doesn't satisfy any recurrence relation. 3n+ 1 for every 040 = 1₂ 4 = 5am-1-1 for every n ≥ 1. O ap=1, an = a1 + 3 for every integer n ≥ 1. ○a=1, an=2(an-1 +1) for every n > 1.

Does the infinite series below converge or diverge? If it converges, what is the sum? 10+2+2/5+2/25+... It converges; the sum is 12.5. it converges; the sum is 15. lt converges; the sum is 25. lt diverges.

A wildlife refuge currently has 100 deer in it. A local wildlife society decides to add an additional 2 deer each month. It is already known that the deer population is growing 12% per year. The size of the population is given by the recursively defined sequence P0=100 Pn=1.01p n-1 + 2 How many deer are approximately in the wildlife refuge at the end of the second month? That is, what is p 2? 1060 deer 109 deer 106 deer 111 deer

Find the nth term and the indicated term of the arithmetic sequence whose initial term, a, and common difference, d, are given. a=-9; d = 4 an=?;a6=? an-13+4n; a6 = 43 an=-13-4n; a6 = 11 an=-13+ 4n; a6 = 11 an=-9 +4n; a6 = 11

Examine each of the series below. Odd numbers: 1+3+5+...+149 Even numbers: 2+4+6+... + 150 a. Write an expression for the nth term of each series. b. What is the sum of each series?

A runner's journal shows the number of miles run each day for a week: Sunday: 3 miles Monday: 5 miles Tuesday: 7 miles Wednesday: 9 miles Thursday: 11 miles Friday: 13 miles Saturday: Based on the arithmetic sequence how many miles are planned for Saturday? A) 38 miles B) 53 miles D 11 miles I 15 miles

Kathleen correctly determines four arithmetic means between -7 and 14. What values does Kathleen determine? O Kathleen determines the four arithmetic means to be -2.8, 1.6, 5.8, and 10. O Kathleen determines the four arithmetic means to be -2.8, 1.4.5.6, and 9.4. O Kathleen determines the four arithmetic means to be 1.6, 5.6, and 9.8. O Kathleen determines the four arithmetic means to be -2.8.1.4, 5.6, and 9.8.

Amit saves certain amount every month in a specific way .In the first month he saves rs 200, in the second month rs 250, In the third month rs 300 and so on. How much will his total savings be in 17 months?

Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. (If the quantity diverges, enter DIVERGES.) an = (n − 2)! /n!

Suppose we want to form five-digit numbers using the set of digits {0,1,2,3). For example, 30133 and 22301 are such numbers, but 03731 is not. How many of these numbers are multiples of 10?

1. The first row of seating in an outdoor amphitheater contains 26 seats, the second row contains 29 seats, the third row contains 32 seats, and so on. (a) Explain why the number of seats in each row can is an arithmetic sequence. (b) Determine the general term of the sequence, an. (c) Suppose there are 20 rows. Write the partial sum using the sigma notation for the total number of seats. (d) What is the total seating capacity of the theater?

Write an expression that gives the requested sum. The sum of the first 22 terms of the geometric sequence 8, 16/ 3, 32/9, ...

Write an arithmetic sequence that has three arithmetic means between 155 and 215. 155, 165, 175, 185, 215 155, 200, 185, 170, 215 155, 175, 195, 205, 215 155, 170, 185, 200, 215

Find the geometric means in the following sequence. -14, ?, ?, ?, ?, -235,298 98, 686, 4,802, 33,614 -1,960, -2,940, -3,920, -4,900 -98,-686, -4,802, -33,614 -686, -4,802, -33,614, -235,313

Write a recursive formula for finding the nth term of each geometric sequence. 5, 20, 80, ... a₁ = 20, a₁ = 4an - 1 a₁ = 80, an=4an - 1 a₁ = 5, an=4an - 2 a₁ = 5, an=4an - 1

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. 1+ 1/9 +1/81+... Select the correct choice below and fill in any answer boxes within your choice. A. The series converges. The sum of the series is (Type an integer or a simplified fraction.) B. The series diverges.

The sequence 6, 18, 54, 162, ... shows the number of pushups Kendall did each week, starting with her first week of exercising. (e) What is the recursive rule for the sequence? (f) What is the iterative rule for the sequence?

You visit the Grand Canyon and drop a rock off the edge of a cliff. The distance the rock will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence. What is the total distance, in feet, the object will fall in 6 seconds? 576 96 176 288 1152

Write the first six terms of the sequence whose nth term is (-1)^/(2n + 2). a1 = a2 = a3 = a4 = a5 = a6 =

Maria Louisa looks at a sequence of shapes and records the number of vertices in each of the shapes in the sequence. The first shape has 7 vertices, the second shape has 10 vertices, and the third shape has 13 vertices. If Maria Louisa assumes that the number of vertices in the sequence of shapes follows the same pattern, how many vertices would she expect the 28 shape to have? A 77 B 85 C) 88 D 91 E 94

Consider the sequence 71.65.59.53.47.41..... The common difference of this sequence is The explicit formula for the sequence is An = Which value is correct for the term below? Select one A18 = -25 A18 = -31 A18 = -37

Find the sum. 9+27+45+...+(18n - 9)

In Exercises 10-6, find the common difference for each arithmetic sequence. 1. 2,6,10,14,... 2. 3,8,13,18,... 3. -7, -2,3,8,... 4. -10,-4,2,8,... 5. 714, 711, 708, 705,... 6. 611, 606, 601, 596,...

Evaluate the geometric series or state that it diverges. 1/8+ 3/64+ 9/512 +27/4096 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The series evaluates to B. The series diverges.

Find the sum. 1/6+5/6+5/6^2+5/6^3+...+5/6^n-1 Complete the sum of the sequence.

After knee surgery, your trainer tells you to return to your jogging program slowly. She suggests jogging for 12 minutes each day for the first week. Each week thereafter, she suggests that you increase that time by 6 minutes per day. How many weeks will it be before you are up to jogging 60 minutes per day? 10 9 8 12 2

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 5th term is 6; 20th term is 66 What is the first term of the sequence? - 10 What is the common difference? 4 What is the recursive formula for the sequence? a₁ = -10 an =an-1 What is the formula for the nth term of the sequence? an +4 .....

Find the indicated term for the given arithmetic sequence. The 100th term of 8,16,24,... a100=

Find the nth term of the arithmetic sequence whose initial term is a1 and common difference is d. What is the forty-fifth term? a₁ = 3; d = -5 Enter the formula for the nth term of this arithmetic series. an = (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Given a x³ + y³ + z³ subject to the constraints x + y + z = 1 and x² + y² + z² = 1 a) Find the Hamiltonian equation b) Find the solution of Hamiltonian equation x, y, z c) Determine maximum and minimum values.

You start a chain email and send it to six friends. The next day, each of your friends forwards the email to six people. The process continues for a few days. a. Write a function that represents the number of people who have received the email after n days. an b. After how many days will 1296 people have received the email? days

Last year, 150 cases were reported of a new infectious disease. It has been predicted that the number will double every year. How many cases will be reported in the ninth year? 76,800 38,400 166 256

On his first day as a telemarketer, Marshall made 24 calls. His goal was to make 5 more calls on each successive day than he had made the day before. If Marshall met, but did not exceed, his goal, how many calls had he made in all after spending exactly 20 days making calls as a telemarketer? A. 670 B. 690 C. 974 D. 1,430 E. 1,530

Makya was conducting a physics experiment. He rolled a ball down a ramp and calculated the distance covered by the ball at different times. The ball rolled a distance of 1 foot during the first second, 3 feet during the next second, and so on. If the distances the ball rolled down the ramp each second form an arithmetic sequence, determine the distance the ball rolled down during the fifteenth second.

A music concert is organized at a memorial auditorium. The first row of the auditorium has 16 seats, the second row has 24 seats, the third row has 32 seats, and so on, increasing by 8 seats each row for a total of 50 rows. Find the number of people that can be accommodated in the sixteenth row.