In a Certain Sequence, the Term ​​\(x_n\) is Given by the Formula GMAT Problem Solving

Question:  In a certain sequence, the term \(x_n\) is given by the formula \(x_n = 2*x_{n-1}- \frac{1}{2} *x_{n-2}\)  for all n ≥ 2. If \(x_0\) = 3 and \(x_1\) = 2, what is the value of \(x_3\)?

(A) 2.5
(B) 3.125
(C) 4
(D) 5
(E) 6.75

Correct Answer: C

Solution and Explanation:

Approach Solution 1:

For all n > = 2, it is stated that \(x_n = 2*x_{n-1}- 12 *x_{n-2}\) exists. This is a recursive formula, which means that in order to calculate the upcoming terms, we must first know the preceding terms. For instance, in order to determine X(2), we must also know \(x_1\) and \(x_0\) because \(x_2\) equals 2 * \(x_1\) - 12 * \(x_0\) (0).

The values \(x_1\) = 2 and \(x_0\) = 3 are presented. Therefore, \(x_2\) would be calculated as follows when n is 2:

\(x_2 = 2*x_1- \frac{1}{2} *x_0(0)\)

\(x_2\) = 2 x 2 - 1.2 x 3

\(x_2\) = 4 – 1.5

\(x_2\) = 2.5

We can now calculate the value of X. (3). In this instance, n = 3 and X(1), X(2) are both 2. In the recursive formula provided in the question stem, we enter these values:

\(x_3\) = 2 * \(x_2\) – ½ * \(x_1\) (1)

\(x_3\) = 2 * 2.5 – ½ * 2

\(x_3\) = 5 – 1

\(x_3\) = 4

C is the correct answer.

Approach Solution 2:

To get the values of the terms in the series beginning with \(x_2\), we can use the formula \(x_n = 2*x_{n-1}- 12 *x_{n-2}\). Hence:

\(x_2\)=2∗\(x_1\)−1 / 2∗\(x_0\) = 2 ∗ 2 − ½ ∗ 3 =5/2

\(x_3\)= 2 ∗ \(x_2\) − ½ ∗ \(x_1\)= 2 ∗ 5/2 −1/ 2 ∗ 2 = 4

C is the correct choice.

“In a certain sequence, the term xn is given by the formula" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

Suggested GMAT Problem Solving Samples

• What is the Value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2 GMAT Problem Solving
• A Jar Full of Whisky Contains 40% Alcohol GMAT Problem Solving
• X is Older Than Y, Z Is Younger Than W And V is Older Than Y. Is Z Younger Than X?
• A Flagstaff 17.5 m High Casts a Shadow of Length 40.25 m
• Frances Can Complete a Job in 12 hours, and Joan Can Complete the Same Job in 8 Hours GMAT Problem Solving
• Running at the Same Constant Rate, 6 Identical Machines can Produce a Total of 270 Bottles Per Minute. GMAT Problem Solving
• If abc ≠ 0 is (a/b)/c = a/(b/c) GMAT Problem Solving
• If 0 < a < b < c, Which of the Following Statements Must be True? GMAT Problem Solving
• Find The Value Of x
• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours GMAT Problem Solving
• The Value of k if the Sum of Consecutive Odd Integers From 1 to k Equals 441 GMAT Problem Solving
• A Circle with a Radius R is Inscribed into a Square with a Side K. GMAT Problem Solving
• Machine A can do a Certain Job in 12 Days Working 2 Full Shifts GMAT Problem Solving
• The square of 5√252 =? GMAT Problem Solving
• After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving
• If 12 Ounces of a Strong Vinegar Solution are Diluted with 50 Ounces of Water to Form a Three-Percent Vinegar Solution GMAT Problem Solving
• A Contractor Estimated that his 10-Man Crew Could Complete the Construction in 110 Days if There Was no Rain. GMAT Problem Solving
• At a Dog Competition, a Dog is Awarded 10 Points if it Runs Through 4 Pipes, Makes 10 Jumps, and Walks on 2 Beams. GMAT Problem Solving
• The Population of the Bacteria Colony Doubles Every Day GMAT Problem Solving
• If tu=xytu=xyand ty=uxty=ux Where t, u, x, and y are Non-Zero Integers GMAT Problem Solving
• A Farm has Chickens, Cows and Sheep. The Number of Chickens and Cows Combined is 3 Times the Number of Sheep. GMAT Problem Solving
• In how Many Different Ways Can a Group of 8 People be Divided into 4 Teams of 2 People Each? GMAT Problem Solving
• If m is Three Times n, and if 2n + 3 is 20% of 25, What is the value of m? GMAT Problem Solving
• If Ben Were to Lose the Championship, Mike would be the Winner GMAT Problem Solving
• A Train Travelling at a Certain Constant Speed takes 30 seconds GMAT Problem Solving
• A Conference Room is Equipped with a Total of 45 Metal or Wooden Chairs.GMAT Problem Solving
• How many litres of a 90% solution of concentrated acid needs to be mixed with a 75% solution GMAT problem solving
• Jug Contains Water And Orange Juice In The Ratio 5:7 . Another Jug Contains Water And Orange J GMAT problem solving
• The number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated GMAT problem solving
• A train travels from Albany to Syracuse, a distance of 120 miles, at an average rate of 50 miles per hour GMAT problem solving
• The hexagon ABCDEF is regular. That means all its sides are the same length and all its interior angles are the same size. GMAT problem solving
• y varies directly as x and when x = 6, y = 24. What is the value of y, when x = 5? GMAT problem solving
• If k is an Integer and 2 < k < 7, for How Many Different Values of k is There a Triangle With Sides of Lengths 2, 7, and k? GMAT problem solving
• How many factors does 36^2 have? GMAT problem solving
• A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively GMAT problem solving
• Walking at 6/7 th of his usual speed, a man is 25 min too late GMAT problem solving
• An Inlet Pipe can Fill in an Empty Cistern in 30 minutes Whereas a leak in the Bottom of the Cistern can Empty a Filled Tank in 40 minutes GMAT problem solving
• A milkman cheats his customers by adding water to the milk he sells GMAT problem solving
• if 80 lamps can be lighted, 5 hours per day for 10 days for $21.25, then the number of lamps, GMAT problem solving
• Few of the corporate contributions to the earthquake relief fund, aside from Pterocom GMAT problem solving
• The product of the first 10 prime numbers is closest to which of the following? GMAT problem solving
• There is a 120 liter mixture of alcohol and water. The ratio of alcohol to water is 7 : 5 GMAT problem solving
• An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles GMAT problem solving
• A Furniture Store Sells Only Two Models of Desks, Model A and Model B. The Selling Price of Model A is $120 GMAT problem solving
• A contractor combined x tons of a gravel mixture that contained 10 percent gravel G GMAT problem solving
• There are 100 Apples in a Bag of which 98% are Green and Rest are Red GMAT problem solving
• The Smallest of Six Consecutive Odd Integers Whose Average (arithmetic mean) is x + 2 GMAT Problem Solving
• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours. GMAT Problem Solving
• How Many Litres of a 90% Solution of Concentrated Acid Needs to be Mixed with a 75% Solution GMAT Problem Solving GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200! GMAT Problem Solving