2.Progression Flashcards
What is an Arithmetic Progression (AP)?
An Arithmetic Progression is a sequence of numbers in which the difference between consecutive terms is constant.
What is the formula for the nth term of an Arithmetic Progression?
The formula for the nth term of an AP is: a + (n - 1) * d, where ‘a’ is the first term and ‘d’ is the common difference.
What is the formula for the sum of the first ‘n’ terms of an Arithmetic Progression?
The sum of the first ‘n’ terms of an AP is given by the formula: (n/2) * (2a + (n - 1) * d), where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ is the number of terms.
What is a Geometric Progression (GP)?
A Geometric Progression is a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the common ratio.
What is the formula for the nth term of a Geometric Progression?
The formula for the nth term of a GP is: a * r^(n - 1), where ‘a’ is the first term and ‘r’ is the common ratio.
What is the formula for the sum of the first ‘n’ terms of a Geometric Progression?
The sum of the first ‘n’ terms of a GP is given by the formula: (a * (1 - r^n))/(1 - r), where ‘a’ is the first term, ‘r’ is the common ratio, and ‘n’ is the number of terms.
What is a Harmonic Progression (HP)?
A Harmonic Progression is a sequence of numbers in which the reciprocal of each term forms an arithmetic progression.
What is the formula for the nth term of a Harmonic Progression?
The formula for the nth term of an HP is: 1/(a + (n - 1) * d), where ‘a’ is the first term and ‘d’ is the common difference (reciprocal of the AP formed by the terms).
What is the formula for the sum of the first ‘n’ terms of a Harmonic Progression?
The sum of the first ‘n’ terms of an HP is given by the formula: (n * (2a + (n - 1) * d))/(2(a + (n - 1) * d)), where ‘a’ is the first term, ‘d’ is the common difference, and ‘n’ is the number of terms.
What is the formula for the Geometric Mean (GM)?
The formula for the Geometric Mean is: GM = (a1 * a2 * a3 * … * an)^(1/n), where a1, a2, …, an are the terms of the sequence.
What is the sum formula for a Geometric Progression (GP) when the common ratio (r) is greater than 1?
Sum of GP (r > 1): S = (a * r^n - 1)/(r - 1), where ‘a’ is the first term, ‘r’ is the common ratio, and ‘n’ is the number of terms.
What is the sum formula for a Geometric Progression (GP) when the common ratio (r) is less than 1?
Sum of GP (r < 1): S = (a * (1 - r^n))/(1 - r), where ‘a’ is the first term, ‘r’ is the common ratio, and ‘n’ is the number of terms.
What is the formula for the sum of an infinite Geometric Progression (GP)?
Sum of infinite GP: S = (a)/(1 - r), where ‘a’ is the first term and ‘r’ is the common ratio.
What is the formula for the Harmonic Mean (HM)?
The formula for the Harmonic Mean is: HM = n/(1/a1 + 1/a2 + 1/a3 + … + 1/an), where a1, a2, …, an are the terms of the sequence.
What is the formula for the sum of an infinite Harmonic Progression (HP)?
Sum of infinite HP: S = a/(1 - d), where ‘a’ is the first term and ‘d’ is the common difference (reciprocal of the AP formed by the terms).
