Mathematics ยท 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

Determine if the series $\sum_{n=1}^\infty \frac{n^2+1}{n^3+1}$ 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 $\sum_{n=1}^\infty \frac{n^2+1}{n^3+1}$ is convergent because it satisfies the limit comparison test. Comparing it to the series $\sum_{n=1}^\infty \frac{1}{n}$, which is convergent, we have $\lim_{n\to\infty} \frac{\frac{n^2+1}{n^3+1}}{\frac{1}{n}} = \lim_{n\to\infty} \frac{n^3+n}{n^3+1} = 1$. Therefore, $\sum_{n=1}^\infty \frac{n^2+1}{n^3+1}$ is also convergent.

Multiple choice

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

  1. $\frac{1}{2}$
  2. $\frac{3}{2}$

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

The series $\sum_{n=1}^\infty \frac{1}{n(n+2)}$ is a telescoping series. Rewriting the terms as $\frac{1}{n} - \frac{1}{n+2}$, we have $\sum_{n=1}^\infty \frac{1}{n(n+2)} = \sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n+2}\right) = \left(1 - \frac{1}{3}\right) + \left(\frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + \cdots = \frac{1}{2}$.

Multiple choice

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

  1. Convergent

  2. Divergent

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

The series $\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}$ is divergent because it is an alternating series that does not satisfy the alternating series test. The terms of the series do not decrease in absolute value, and the limit of the terms as $n\to\infty$ is not 0. Therefore, the series diverges.

Multiple choice

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

  1. $\frac{1}{2}$
  2. $\frac{1}{3}$
  3. $\frac{1}{4}$
  4. $\frac{1}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The series $\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$ is a telescoping series. Rewriting the terms as $\frac{1}{n(n+1)(n+2)} = \frac{1}{2} \left(\frac{1}{n(n+1)} - \frac{1}{(n+1)(n+2)}\right)$, we have $\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)} = \frac{1}{2} \left[\left(\frac{1}{1\cdot2} - \frac{1}{2\cdot3}\right) + \left(\frac{1}{2\cdot3} - \frac{1}{3\cdot4}\right) + \left(\frac{1}{3\cdot4} - \frac{1}{4\cdot5}\right) + \cdots\right] = \frac{1}{4}$.

Multiple choice

Find the sum of the series $\sum_{n=1}^\infty \frac{2^n}{5^n}$.

  1. $\frac{2}{3}$
  2. $\frac{3}{5}$
  3. $\frac{4}{5}$
  4. $\frac{5}{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The series $\sum_{n=1}^\infty \frac{2^n}{5^n}$ is a geometric series with first term $a = \frac{2}{5}$ and common ratio $r = \frac{2}{5}$. The sum of a geometric series is given by $S = \frac{a}{1-r}$. Substituting the values of $a$ and $r$, we get $S = \frac{\frac{2}{5}}{1-\frac{2}{5}} = \frac{2}{3}$.

Multiple choice

Determine whether the series $\sum_{n=1}^\infty \frac{n^2+2n+1}{n^3+3n^2+2n}$ 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 $\sum_{n=1}^\infty \frac{n^2+2n+1}{n^3+3n^2+2n}$ is convergent because it satisfies the limit comparison test. Comparing it to the series $\sum_{n=1}^\infty \frac{1}{n}$, which is convergent, we have $\lim_{n\to\infty} \frac{\frac{n^2+2n+1}{n^3+3n^2+2n}}{\frac{1}{n}} = \lim_{n\to\infty} \frac{n^3+3n^2+2n}{n^3+3n^2+2n} = 1$. Therefore, $\sum_{n=1}^\infty \frac{n^2+2n+1}{n^3+3n^2+2n}$ is also convergent.

Multiple choice

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

  1. $\frac{1}{2}$
  2. $\frac{2}{3}$
  3. $\frac{3}{4}$
  4. $\frac{4}{5}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The series $\sum_{n=1}^\infty \frac{1}{n(n+2)}$ is a telescoping series. Rewriting the terms as $\frac{1}{n(n+2)} = \frac{1}{2} \left(\frac{1}{n} - \frac{1}{n+2}\right)$, we have $\sum_{n=1}^\infty \frac{1}{n(n+2)} = \frac{1}{2} \left[\left(\frac{1}{1\cdot3} - \frac{1}{3\cdot5}\right) + \left(\frac{1}{3\cdot5} - \frac{1}{5\cdot7}\right) + \left(\frac{1}{5\cdot7} - \frac{1}{7\cdot9}\right) + \cdots\right] = \frac{2}{3}$.

Multiple choice

What is the mathematical formula for calculating the progressions?

  1. $$P = \frac{1}{\sqrt{\frac{G M}{a^3}}}$$
  2. $$P = \frac{1}{\sqrt{\frac{G m}{a^3}}}$$
  3. $$P = \frac{2\pi}{\sqrt{\frac{G M}{a^3}}}$$
  4. $$P = \frac{2\pi}{\sqrt{\frac{G m}{a^3}}}$$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The progressions are the time it takes for a planet to move one degree in longitude. The formula for calculating the progressions is $$P = \frac{2\pi}{\sqrt{\frac{G M}{a^3}}}$$, where $$G$$ is the gravitational constant, $$M$$ is the mass of the Sun, and $$a$$ is the semi-major axis of the planet's orbit.

Multiple choice

What is the formula for finding the sum of the first (n) natural numbers?

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

The formula for the sum of the first (n) natural numbers is (\frac{n(n+1)}{2}).

Multiple choice

What is the formula for the Nilakantha series?

  1. $$\pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4 \cdot 5 \cdot 6} + \frac{4}{6 \cdot 7 \cdot 8} - \cdots$$
  2. $$\pi = 4 - \frac{4}{2 \cdot 3} + \frac{4}{4 \cdot 5} - \frac{4}{6 \cdot 7} + \cdots$$
  3. $$\pi = 3 + \frac{4}{2 \cdot 3} + \frac{4}{4 \cdot 5} + \frac{4}{6 \cdot 7} + \cdots$$
  4. $$\pi = 4 - \frac{4}{2 \cdot 3 \cdot 4} + \frac{4}{4 \cdot 5 \cdot 6} - \frac{4}{6 \cdot 7 \cdot 8} + \cdots$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the Nilakantha series is $$\pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4 \cdot 5 \cdot 6} + \frac{4}{6 \cdot 7 \cdot 8} - \cdots$$.

Multiple choice

What is the formula for the Nilakantha series?

  1. $$\pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4 \cdot 5 \cdot 6} + \frac{4}{6 \cdot 7 \cdot 8} - \cdots$$
  2. $$\pi = 4 - \frac{4}{2 \cdot 3} + \frac{4}{4 \cdot 5} - \frac{4}{6 \cdot 7} + \cdots$$
  3. $$\pi = 3 + \frac{4}{2 \cdot 3} + \frac{4}{4 \cdot 5} + \frac{4}{6 \cdot 7} + \cdots$$
  4. $$\pi = 4 - \frac{4}{2 \cdot 3 \cdot 4} + \frac{4}{4 \cdot 5 \cdot 6} - \frac{4}{6 \cdot 7 \cdot 8} + \cdots$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the Nilakantha series is $$\pi = 3 + \frac{4}{2 \cdot 3 \cdot 4} - \frac{4}{4 \cdot 5 \cdot 6} + \frac{4}{6 \cdot 7 \cdot 8} - \cdots$$. This series converges very quickly, and it can be used to calculate the value of pi to a high degree of accuracy.

Multiple choice

What is the formula for the Nilakantha series for finding the sum of an infinite series?

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

The formula for the Nilakantha series for finding the sum of an infinite series is $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$. This series converges very quickly, and it can be used to find the sum of many different infinite series.

Multiple choice

What is the sum of the infinite series $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$?

  1. 0

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

The sum of the infinite series $$S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$$ is $$\frac{1}{2}$$. This can be shown using the Nilakantha series.

Multiple choice

What is the formula for calculating the Simpson's Index?

  1. D = 1 - \sum_{i=1}^{S} p_i^2

  2. D = \sum_{i=1}^{S} p_i^2

  3. D = \frac{1}{\sum_{i=1}^{S} p_i^2}

  4. D = \frac{1}{S \sum_{i=1}^{S} p_i^2}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Simpson's Index is calculated by summing the squared proportions of each species in the community and subtracting the result from 1.

Multiple choice

Find the generating function for the sequence (1, 1, 2, 3, 5, 8, \dots), where each term is the sum of the previous two terms.

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

The generating function for the sequence (1, 1, 2, 3, 5, 8, \dots) is (\frac{x}{1-x-x^2}) because the characteristic equation of the recurrence relation (a_n = a_{n-1} + a_{n-2}) is (x^2 + x + 1 = 0), and the roots of this equation are (\frac{-1 \pm \sqrt{-3}}{2}).