Mathematics · Quantitative Aptitude

Sequences and Series

230 Questions

Sequences and series involve ordered lists of numbers and the sum of their terms. The questions primarily test knowledge of arithmetic progressions, geometric progressions, and infinite series. It is an essential part of quantitative aptitude that requires strong pattern recognition skills.

Arithmetic progressionGeometric progressionInfinite seriesSum of termsNumber sequences

Sequences and Series Questions

Multiple choice

Use the Möbius Inversion Formula to find a formula for the sum of the Euler phi function over the divisors of an integer ( n ).

  1. \( \sum_{d|n} \phi(d) = n \)
  2. \( \sum_{d|n} \phi(d) = \phi(n) \)
  3. \( \sum_{d|n} \phi(d) = \sigma(n) \)
  4. \( \sum_{d|n} \phi(d) = \mu(n) \)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the Möbius Inversion Formula with ( f(n) = \phi(n) ) and ( g(n) = \sum_{d|n} \mu(d) \phi(d) ), we get ( \phi(n) = \sum_{d|n} \mu(d) g(\frac{n}{d}) ). Since ( g(\frac{n}{d}) = \phi(\frac{n}{d}) ) for all ( d | n ), we have ( \sum_{d|n} \phi(d) = \phi(n) ).

Multiple choice

Use the Möbius Inversion Formula to find a formula for the sum of the Möbius function over the divisors of an integer ( n ) that are relatively prime to ( n ).

  1. \( \sum_{d|n} \mu(d) = 1 \)
  2. \( \sum_{d|n} \mu(d) = n \)
  3. \( \sum_{d|n} \mu(d) = \phi(n) \)
  4. \( \sum_{d|n} \mu(d) = \sigma(n) \)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If ( d ) is a divisor of ( n ) that is relatively prime to ( n ), then ( \frac{n}{d} ) is also a divisor of ( n ) that is relatively prime to ( n ). Therefore, ( \mu(d) = \mu(\frac{n}{d}) ). Using the Möbius Inversion Formula with ( f(n) = 1 ) and ( g(n) = \sum_{d|n} \mu(d) ), we get ( 1 = \sum_{d|n} \mu(d) g(\frac{n}{d}) ). Since ( g(\frac{n}{d}) = \phi(\frac{n}{d}) ) for all ( d | n ) that are relatively prime to ( n ), we have ( \sum_{d|n} \mu(d) = \phi(n) ).

Multiple choice

What is the generating function for the sequence {1, 1, 2, 3, 5, 8, 13, ...}, where each term is the sum of the two previous terms?

  1. G(x) = 1 / (1 - x - x^2)

  2. G(x) = 1 / (1 + x + x^2)

  3. G(x) = 1 / (1 - x + x^2)

  4. G(x) = 1 / (1 + x - x^2)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The generating function for the sequence {1, 1, 2, 3, 5, 8, 13, ...} is G(x) = 1 / (1 - x - x^2), which can be obtained using the formula for the generating function of a Fibonacci-like sequence.

Multiple choice

What is the 10th term in the Fibonacci sequence?

  1. 34

  2. 55

  3. 89

  4. 144
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The Fibonacci sequence is a sequence of numbers where each number is the sum of the two preceding numbers. The first two numbers in the sequence are 0 and 1. The next number is 0 + 1 = 1, the next number is 1 + 1 = 2, and so on. The 10th term in the Fibonacci sequence is 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144 = 55.

Multiple choice

What is the formula for the present value of a single sum?

  1. PV = FV / (1 + r)^n

  2. PV = FV * (1 + r)^n

  3. PV = FV / (1 - r)^n

  4. PV = FV * (1 - r)^n
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The present value of a single sum is calculated by dividing the future value by the factor (1 + r)^n, where r is the interest rate and n is the number of years.

Multiple choice

What is the formula for the future value of a single sum?

  1. FV = PV * (1 + r)^n

  2. FV = PV / (1 + r)^n

  3. FV = PV * (1 - r)^n

  4. FV = PV / (1 - r)^n
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The future value of a single sum is calculated by multiplying the present value by the factor (1 + r)^n, where r is the interest rate and n is the number of years.

Multiple choice

What is the formula for calculating the Future Value (FV) of a single sum?

  1. FV = PV * (1 + r)^n

  2. FV = PV * (1 - r)^n

  3. FV = PV / (1 + r)^n

  4. FV = PV / (1 - r)^n
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for calculating the Future Value (FV) of a single sum is FV = PV * (1 + r)^n, where PV is the Present Value, r is the interest rate, and n is the number of periods.

Multiple choice

What is the formula for calculating the Present Value (PV) of a single sum?

  1. PV = FV / (1 + r)^n

  2. PV = FV * (1 + r)^n

  3. PV = FV * (1 - r)^n

  4. PV = FV / (1 - r)^n
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for calculating the Present Value (PV) of a single sum is PV = FV / (1 + r)^n, where FV is the Future Value, r is the interest rate, and n is the number of periods.

Multiple choice

What is the name of the theorem that states that the sum of the squares of the first n natural numbers is equal to n(n+1)(2n+1)/6?

  1. Pythagorean theorem

  2. Euler's theorem

  3. Fermat's Last Theorem

  4. Gauss's formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Gauss's formula states that the sum of the squares of the first n natural numbers is equal to n(n+1)(2n+1)/6.

Multiple choice

Which theorem states that the sum of the squares of the first (n) natural numbers is (\frac{n(n+1)(2n+1)}{6})?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Angle Sum Theorem

  4. Gauss's Formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Gauss's Formula states that the sum of the squares of the first (n) natural numbers is (\frac{n(n+1)(2n+1)}{6}).

Multiple choice

What is the formula for finding the sum of the first n natural numbers according to Virasena?

  1. $$S_n = \frac{n(n+1)}{2}$$
  2. $$S_n = \frac{n(n-1)}{2}$$
  3. $$S_n = \frac{n(n+2)}{2}$$
  4. $$S_n = \frac{n(n-2)}{2}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to Virasena, the formula for finding the sum of the first n natural numbers is $$S_n = \frac{n(n+1)}{2}$$

Multiple choice

The equation (\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}) is known as:

  1. Basel problem

  2. Riemann hypothesis

  3. Goldbach's conjecture

  4. Catalan's conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation (\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}) is known as the Basel problem, which was first solved by Leonhard Euler in the 18th century.

Multiple choice

Determine whether the series $\sum_{n=1}^\infty \frac{(-1)^n}{n^2}$ is convergent or divergent.

  1. Convergent

  2. Divergent

  3. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The series $\sum_{n=1}^\infty \frac{(-1)^n}{n^2}$ is convergent because it is an alternating series that satisfies the alternating series test. The terms of the series decrease in absolute value, and the limit of the terms as $n\to\infty$ is 0. Therefore, the series converges.

Multiple choice

Find the sum of the series $\sum_{n=1}^\infty \frac{1}{2^n}$.

  1. 1

  2. 2
  3. $\frac{1}{2}$
  4. $\frac{2}{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The series $\sum_{n=1}^\infty \frac{1}{2^n}$ is a geometric series with first term $a = \frac{1}{2}$ and common ratio $r = \frac{1}{2}$. The sum of a geometric series is given by $S = \frac{a}{1-r}$. Substituting the values of $a$ and $r$, we get $S = \frac{\frac{1}{2}}{1-\frac{1}{2}} = 1$. Therefore, the sum of the series is 1.

Multiple choice

Which of the following series is absolutely convergent?

  1. $\sum_{n=1}^\infty (-1)^n \frac{1}{n}$
  2. $\sum_{n=1}^\infty \frac{n}{n+1}$
  3. $\sum_{n=1}^\infty \frac{1}{n^2}$
  4. $\sum_{n=1}^\infty \frac{(-1)^n n}{n+1}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A series is absolutely convergent if the series of absolute values of its terms is convergent. The series $\sum_{n=1}^\infty \frac{1}{n^2}$ is absolutely convergent because the series $\sum_{n=1}^\infty \frac{1}{n^2}$ is convergent. The series $\sum_{n=1}^\infty \frac{n}{n+1}$ is not absolutely convergent because the series $\sum_{n=1}^\infty \frac{n}{n+1}$ is divergent.