The nth term of an increasing sequence S is given by S_{n} = S_{n-1} + S_{n-2} for n > 2 and the nth term of a sequence S’ is given by S’_{n} = S’_{n-1} - S’_{n-2 }for n > 2. If S_{5} = S’_{5}, what is the average (arithmetic mean) of S_{2} and S’_{2}?
(1) The difference between the fourth term and the second term of sequence S is 14.
(2) The sum of the fourth term and the second term of sequence S’ is 14.
Steps 1 & 2: Understand Question and Draw Inferences
Thus we need to find the value of S_{3} to find the average of S_{2} andS′_{2}
Step 3: Analyze Statement 1 independently
(1) The difference between the fourth term and the second term of sequence S is 14
As we know the value of S_{3}, the statement is sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The sum of the fourth term and the second term of sequence S’ is 14.
Does not tell us anything about the value of S3, the statement is insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step -3, this step is not required.
Answer: A
If the sum of the first 30 positive odd integers is k, what is the sum of first 30 non-negative even integers?
Given
To Find: 0 + 2+ 4………30 *2 -2 = ?
Approach
Working Out
Answer: B
240 students are to be arranged in n rows in the assembly hall of a school. If the first row is the closest to the stage and each subsequent row has 10 more students than the row ahead of it, what is the value of n?
(1) There are 45 students in the 4^{th} row from the stage.
(2) The number of students in the n^{th} row is 10 less than 5 times the number of students in the first row.
Steps 1 & 2: Understand Question and Draw Inferences
and that in the last row be a_{n}
(average of the 1^{st} and the nth term is the average of all the terms of the sequence)
To Find: Unique value of n
Step 3: Analyze Statement 1 independently
(1) There are 45 students in the 4^{th} row from the stage.
Substituting the value of a_{1} in (1), we have
As n cannot be negative, n = 6.
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The number of students in the n^{th} row is 10 less than 5 times the number of students in the first row.
Putting the value of in (1), we have
As n has to be an integer, n = 6.
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
An increasing sequence P consists of 10 distinct integers. How many integers of the sequence are less than 16?
(1) The difference between any two integers of the sequence is divisible by 2 and 3.
(2) If the third term of the sequence P is removed, the magnitude of the product of the terms of the sequence remains unchanged.
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
(1) The difference between any two integers of the sequence is divisible by 2 and 3.
Does not tell us anything about the values of the integers of sequence P.
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) If the third term of the sequence P is removed, the magnitude of the product of the terms of the sequence remains unchanged.
The statement does not tell us anything about the other terms.
Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
(1) Difference between any two integers is of the form (2*3)k = 6k
(2) a_{3} = 1 or -1
Combining both the statements, we cannot say for sure how many terms are less than 16, as we do know the value of 6k.
So, 6k = 6, 12 or 18.
For each case, we will have a different answer.
Insufficient to answer
Answer: E
Steven and Stuart took a job in different companies at the same time. Steven’s salary increased by a fixed amount at the end of every year and Stuart’s salary increased by a fixed percentage at the end of every year. If the increase in the salary of Steven at the end of the third year was equal to the increase in the salary of Stuart at the end of the second year, what was the difference in the salaries of Steven and Stuart when they took the job?
(1) Steven’s salary after 2 years was 20% more than the salary at which he took the job
(2) The increase in the salary of Stuart at the end of the second year was 11% of the salary at which he took the job.
Steps 1 & 2: Understand Question and Draw Inferences
and Stuart’s initial salary be S1
To Find: S_{1}−S_{2}
Step 3: Analyze Statement 1 independently
(1) Steven’s salary after 2 years was 20% more than the salary at which he took the job
However, we do not know anything about the values of S_{2}, x and y.
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The increase in the salary of Stuart at the end of the second year was 11% of the salary at which he took the job.
Insufficient to answer as it does give us the value of S1,S2orx
Step 5: Analyze Both Statements Together (if needed)
Need to know the value of x to answer the question.
Insufficient to answer.
Answer: E
If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?
Given:
To Find:
Approach:
Working out:
Now that we have values of x_{1 } and d. The value of 4^{th} term of the sequence will be
⇒ x1+3(d)=14+3(5)=29
Answer:
Alternate method
Therefore,
Solving Eq(1) and (2) we get
Correct Answer: Option A
A list contains distinct integers a_{1}, a_{2}, …a_{10} arranged in ascending order. If the integers of the list lie between -19 and 19, inclusive such that the distance between any two consecutive integers is equal, is one of the terms of this list equal to zero?
(1) All the integers in the list are divisible by 2
(2) a_{4} = -6
Steps 1 & 2: Understand Question and Draw Inferences
Step 3: Analyze Statement 1 independently
(1) All the integers in the list are divisible by 2
As we do not have a unique answer, the statement is insufficient to answer the question
Step 4: Analyze Statement 2 independently
(2) a_{4} = -6
Hence, cannot say for sure if 0 is a part of the sequence. Insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
(1) d = {2, 4}
(2) a_{4} = -6
Insufficient to answer
Answer: E
An increasing sequence consists of 4 negative integers and 6 positive integers. Is the sum of the sequence positive?
(1) The difference between any two consecutive negative integers is 5 and the difference between any two consecutive positive integers is 2
(2) The first term of the sequence is -16
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is
Step 3: Analyze Statement 1 independently
(1) The difference between any two consecutive negative integers is 5 and the difference between any two consecutive positive integers is 2
Need to know the value of a_{1 }and a_{5 } to know if the sum of the sequence is greater than 0.
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The first term of the sequence is -16
Hence,
Step 5: Analyze Both Statements Together (if needed)
From Statement 1 we got
From Statement 2 we got
Combining both we get:
Now, if we notice carefully a_{5} is a positive integer,
Therefore
Hence
Combining both statements was sufficient to answer the question
Correct Answer: C
An increasing sequence M consists of 5 consecutive positive multiples of a positive integer. What is the remainder when the largest term of the sequence is divided by 2?
(1) The median of the sequence is even.
(2) The second term of the sequence is odd.
Steps 1 & 2: Understand Question and Draw Inferences
To Find:
Step 3: Analyze Statement 1 independently
(1) The median of the sequence is even.
Alternate method
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The second term of the sequence is odd.
Sufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
Mike took 5 mock tests before appearing for the GMAT. In each mock test he scored 10 points more than the previous mock test. If he scored 760 on the GMAT and his average score for the mocks was 720, what is the difference between his last mock score and his GMAT score?
Given
Mike took 5 mock tests
To Find:
Working Out
Average score of mocks = (x + x + 10 + … x + 40) /5 = (5x + 100)/5 = x + 20
(The other way to think about this is, as Mike’s scores in the mocks are in arithmetic sequence, average will be the middle term)
Therefore
Correct Answer: Option B
In the sequence S, the difference between any two consecutive terms is equal. If the sum of the fourth term and the fifth term of the sequence is equal to the seventh term of the sequence, what is the value of the second term of the sequence?
Given
To Find: a + d = ?
Approach
Working Out
Hence, the value of the 2^{nd} term of the sequence = a + d = 0
Answer: B
A city had 1000 migrants in the year 1999. If the number of migrants in the city has doubled every 3 years since 1999, then what was the increase in the population of migrants during the period from 2008 to 2011?
Given
To Find:
Approach
Working Out
Hence, the migrants population increased by 8000 between 2008 and 2011.
Answer: C
An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant, which is also known as the common difference of that arithmetic sequence. The sequence S contains all the terms of two different increasing arithmetic sequences P and Q such that the number of terms in sequence S is equal to the sum of the number of terms in sequences P and Q. If each of the arithmetic sequences P and Q consists of 10 positive integral terms, how many distinct terms does sequence S have?
(1) The least common multiple of the common differences of the sequences P and Q is 6
(2) The third term of the sequence P is equal to the second term of the sequence Q
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Number of distinct terms of sequence S
Step 3: Analyze Statement 1 independently
(1) The least common multiple of the common differences of the sequences P and Q is 6
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The third term of the sequence P is equal to the second term of the sequence Q
Also, we do not know the common difference of the two sequences. Hence the statement is insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Uisng (1) and (2), we cannot calculate the unique common values of the common differences as well as we do not know if P_{3} and Q_{2} are the smallest terms of the sequences P and Q that are common.
Hence, insufficient to answer.
Answer: E
An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant, which is also known as the common difference of that arithmetic sequence. An increasing arithmetic sequence N consists of a set of distinct negative integers and an increasing arithmetic sequence P consists of a set of distinct positive integers. The sequence C contains all the terms of arithmetic sequences N and P such that the number of terms in sequence C is equal to the number of terms in arithmetic sequences N and P. Is sequence C an arithmetic sequence?
(1) The sum of the largest term of the sequence N and the smallest term of the sequence P is zero.
(2) For every integer in sequence N, there exists an integer in sequence P with the same magnitude.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is sequence C an arithmetic sequence?
Step 3: Analyze Statement 1 independently
(1) The sum of the largest term of the sequence N and the smallest term of the sequence P is zero.
However we do not know the value of the common differences of the sequences as well as we do not know the value of y.
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) For every integer in sequence N, there exists an integer in sequence P with the same magnitude.
Also, as we do not know the difference between the largest term of sequence N and the smallest term of sequence P, the statement is insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Combining both the statements tell us that the largest term of sequence N (i.e. x) and the smallest term of sequence P(i.e. y ) have the same magnitude.
However it does not tell us:
Hence, combining the statements is also insufficient to answer.
Answer: E
The sequence S consists of 10 terms: x, x^{2}, x^{3}……x^{10}, where x is a non-zero number. If P is the sum of all the terms in the sequence S, is P3
> 0?
(1) The distance of any term of the sequence S from zero on the number line is not less than 1.
(2) x^{5} = x^{7}
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is P^{3}> 0 ?
So, P > 0, except when -1 < x < 0. Hence, we need to find if -1 < x < 0.
Step 3: Analyze Statement 1 independently
(1) The distance of any term of the sequence S from zero on the number line is not less than 1.
So, we cannot say for sure if P > 0 or not. Insufficient to answer.
Step 4: Analyze Statement 2 independently
So, we cannot say for sure if P > 0. Insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
Combining both the statements, we have x = -1 or 1.
So, we cannot say for sure if P > 0.
Insufficient to answer.
Answer: E
A sequence S consists of 5 distinct positive integers. Are all the integers in the sequence divisible by 5?
(1) The sum of all the integers in the sequence is divisible by 5.
(2) The product of all the integers in the sequence is divisible by 5 but not by 10.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is a_{1},a_{2},a_{3},a_{4},a_{5 } divisible by 5?
Step 3: Analyze Statement 1 independently
(1) The sum of all the integers in the sequence is divisible by 5.
a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=5x
, where x is an integer > 0.
Two cases arise:
Insufficient to answer.
Step 4: Analyze Statement 2 independently
(2) The product of all the integers in the sequence is divisible by 5 but not by 10.
a_{1}∗a_{2}∗a_{3}∗a_{4}∗a_{5}=5y, where y is an integer > 0
For the product of the terms to be divisible by 5, any one or more of the terms should be divisible by 5.
So, 1 or all the 5 terms may be divisible by 5.
Also, as the product of the terms is not divisible by 10, none of the terms is even.
Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
(1) a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=5x
(2) a_{1}∗a_{2}∗a_{3}∗a_{4}∗a_{5}=5y and none of a_{1},a_{2},a_{3},a_{4},a_{5 } is even
Following cases may arise:
Insufficient to answer.
Answer: E
The sequence a_{1}, a_{2},…a_{n} is such that a_{n} = a_{n-1 }+n*d for all n > 1, where d is a positive integer. If a_{3} = 20 and a_{5} = 47, what is the value of a_{7}?
Given
To Find: a_{7}?
Approach
Working Out
Solving (1) and (2), we have a_{1} = 5 and d = 3
Answer: E
For any positive integer z, S_{Z} denotes the sum of the first z positive integers. For example S_{3} = 1+2 + 3 = 6. Which of the following expressions is correct?
Given:
To find: Which of the given 3 expressions is correct?
Approach:
Working Out:
By definition,
Looking at the answer choices, we see that the correct answer is Option E
Ted and Robin start from the same point at 7 AM and drive in opposite directions. Ted doubles his speed after every 90 minutes whereas Robin reduces her speed by half after every 120 minutes. If Ted starts driving at a speed of 10 kilometers/hour and Robin starts driving at a speed of 120 kilometers/hour, how far in kilometers will they be from one another at 1 PM?
Given
To Find: Distance between Ted and Robin at 1 PM
Approach
Distance = Speed * Time
1. We need to find the distance travelled by Ted in 6 hours As Ted doubles his speed after every 1.5 hours, he will travel at 6/1.5=4
2. We need to find the distance travelled by Robin in 6 hours
Working Out
1. Ted
2. Robin
Distance between Ted and Robin at 1 PM = 225 + 420 = 645 kilometers
Answer: D
List A consists of 10 distinct integers arranged in ascending order. Is the difference between the sixth term and the fifth term of list A greater than 5?
(1) The difference between any two integers in list A is a multiple of 5.
(2) The median of the list is an integer.
Given:
To Find: Is a_{6} – a_{5} > 5?
Step 4: Analyse Statement 2 independently
The median of the list is an integer.
Step 5: Analyse Both Statements Together (if needed)
Answer: C
Step 3: Analyse Statement 1 independently
The difference between any two integers in list A is a multiple of 5.
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