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

Find the generating function for the sequence (1, 3, 6, 10, 15, \dots), where each term is the sum of the first (n) positive integers.

  1. \(\frac{x}{(1-x)^3}\)
  2. \(\frac{x}{(1-x)^2}\)
  3. \(\frac{x}{(1-x)}\)
  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, 3, 6, 10, 15, \dots) is (\frac{x}{(1-x)^3}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)}{2}), which is the sum of the first (n) positive integers.

Multiple choice

Find the generating function for the sequence (1, 2, 4, 7, 11, \dots), where each term is the sum of the first (n) odd positive integers.

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

The generating function for the sequence (1, 2, 4, 7, 11, \dots) is (\frac{x}{(1-x)^4}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(2n+1)}{6}), which is the sum of the first (n) odd positive integers.

Multiple choice

Find the generating function for the sequence (1, 3, 6, 10, 15, \dots), where each term is the sum of the first (n) triangular numbers.

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

The generating function for the sequence (1, 3, 6, 10, 15, \dots) is (\frac{x}{(1-x)^4}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(n+2)}{6}), which is the sum of the first (n) triangular numbers.

Multiple choice

Find the generating function for the sequence (1, 4, 10, 20, 35, \dots), where each term is the sum of the first (n) square numbers.

  1. \(\frac{x}{(1-x)^5}\)
  2. \(\frac{x}{(1-x)^4}\)
  3. \(\frac{x}{(1-x)^3}\)
  4. \(\frac{x}{(1-x)^2}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The generating function for the sequence (1, 4, 10, 20, 35, \dots) is (\frac{x}{(1-x)^5}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}), which is the sum of the first (n) square numbers.

Multiple choice

Find the generating function for the sequence (1, 5, 15, 35, 70, \dots), where each term is the sum of the first (n) pentagonal numbers.

  1. \(\frac{x}{(1-x)^6}\)
  2. \(\frac{x}{(1-x)^5}\)
  3. \(\frac{x}{(1-x)^4}\)
  4. \(\frac{x}{(1-x)^3}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The generating function for the sequence (1, 5, 15, 35, 70, \dots) is (\frac{x}{(1-x)^6}) because the coefficient of (x^n) in this generating function is (\frac{n(n+1)(3n-1)}{6}), which is the sum of the first (n) pentagonal numbers.