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# Arithmetic, Geometric And Harmonic Progressions Formulas

**Basic Formulas for AP, GP and HP with definition**

In this Page you will Find Formulas for AP and GP and HP as well as definition also. These are Standard Formulas to Solve any Types of Problems of AP and GP and HP.

**AP stands for Arithmetic progression**

A series of number is termed to be in Arithmetic progression when the difference between two consecutive numbers remain the same.This constant difference is called the common difference.

**GP stands for Geometric progression**

A geometric progression is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant ratio. This constant ratio is called the common ratio.

**HP stands for Harmonic progression**

A harmonic progression is a sequence of numbers in which each term is the reciprocal of an arithmetic progression.

^{2}= AP x HP

### Formulas of Arithmetic Progression (A.P)

In A.P. the next number can be obtained by adding or subtracting the constant number to the previous in the sequence. Therefore, this constant number is known as the common difference(d).

Suppose, if ‘a’ is the first term and ‘d’ be the common difference, then

- nth term of an AP: (a + (n-1)d)
- Arithmetic Mean: Sum of all terms in the AP divided by the number of terms in the AP.
- Sum of ‘n’ terms of an AP: (0.5n*(first term + last term) = 0.5n*[2a + (n-1)d])

### Formulas of Geometric Progression (G.P)

Suppose, if ‘a’ is the first term and ‘r’ be the common ratio, then

- Formula for nth term of GP = a r
^{n-1} - Geometric mean = nth root of the product of ‘n’ terms in the GP.
- Formula to find the geometric mean between two quantities a and b = \sqrt{ab}
- Formula to find the sum of the number of terms in a GP

Let ‘a’ be the first term, ‘r’ be the common ratio and ‘n’ be the number of terms

- if r>1 , then , s_{n} = a \times \frac{r^{n} -1}{r-1}
- if r<1 , then , s_{n} = a \times \frac{1-r^{n}}{1-r}

Sum of infinite terms in a GP(r<1) \frac{a}{1-r}

### Definition of Harmonic Progression (H.P)

Harmonic progression is the series when the reciprocal of the terms are in AP.

For example, \frac{1}{a}, \frac{1}{ (a + d)}, \frac{1}{(a + 2d)}…… are termed as a harmonic progression as a, a + d, a + 2d are in Arithmetic progression.

- First term of a HP is \frac{1}{a}
- There are many Application of Harmonic Progressions.

### Formulas of Harmonic Progression (H.P)

- The nth term in HP is identified by, {a_{n}} = \frac{1}{a+(n-1)d}

- To solve any problem in harmonic progression, a series of AP should be formed first, and then the problem can be solved.

- For two terms ‘a’ and ‘b’,

Harmonic Mean = \frac{(2ab)}{ (a + b)}

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### Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

If GM, AM and HM are the Geometric Mean, Arithmetic Mean and Harmonic Mean of two positive numbers respectively, then

GM^{2} = AM x HM

### Using Formulas of A.P, G.P and H.P in Questions

**Question 1:**The sum of the first 5 terms of a geometric progression is 124. If the common ratio is 2, what is the first term of the progression?

**a) 2**

**b) 4**

**c) 8**

**d) 16**

**Solution:**

**Correct answer: b**

**Question 2:**The 5th term of an arithmetic progression is 17 and the 10th term is 32. What is the common difference?

**a) 3**

**b) 4**

**c) 5**

**d) 6**

**Solution:**

**Correct option :a**

**Question 3:**Find t10

_{ and S10 for the following series: 2, 9,16 }

**a) 65,535**

**b) 55,330**

**c) 45,440**

**d) 35,445**

**Solution:**

**Correct answer : a**

**Question 4:**A number 39 is divided into three parts which are in A.P. and the sum of their squares is 515. Find the largest number.

**(a) 17**

**(b) 15**

**(c) 13**

**(d) 11**

**Solution:**

**Correct answer: 15.**

**Question 5:**The sum of the first and the third term of a geometric progression is 15 and the sum of its first three terms is 21. Find the progression.

**(a) 3,6,12…**

**(b) 12, 6, 3…**

**(c) Both of these**

**(d) None of these**

**Solution:**

Let’s assume the first term of the geometric progression is “a” and the common ratio is “r”.

According to the given information, the sum of the first and third terms is 15, which can be expressed as:

a + ar^2 = 15 —-(1)

The sum of the first three terms is 21, which can be expressed as:

a + ar + ar^2 = 21 —-(2)

To solve these equations, we can subtract equation (1) from equation (2) to eliminate “a”:

(a + ar + ar^2) – (a + ar^2) = 21 – 15

ar = 6

Now, we can substitute this value of “ar” into equation (1):

a + (6/r) = 15

Multiplying both sides by “r” to eliminate the fraction:

ar + 6 = 15r

Rearranging the equation:

15r – ar = 6

r(15 – a) = 6

Since “r” cannot be zero, we can divide both sides by (15 – a):

r = 6 / (15 – a)

Now, let’s analyze the answer choices:

(a) 3, 6, 12…

If we substitute “a = 3” into the equation for “r”, we get:

r = 6 / (15 – 3) = 6/12 = 1/2

However, this value of “r” does not satisfy the given conditions, as the sum of the first and third terms is not 15.

(b) 12, 6, 3…

If we substitute “a = 12” into the equation for “r”, we get:

r = 6 / (15 – 12) = 6/3 = 2

This value of “r” satisfies the given conditions, as the sum of the first and third terms is indeed 15.

Therefore, the correct answer is 12, 6, 3…

**Correct answer: b**

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- Arithmetic Progressions – Questions | Formulas | How to Solve Quickly | Tricks & Shortcuts
- Geometric Progressions – Questions | Formulas | How to Solve Quickly | Tricks & Shortcuts
- Harmonic Progressions – Questions | Formulas | How to Solve Quickly | Tricks & Shortcuts

- Arithmetic Progressions –

Questions |

Formulas |

How to Solve Quickly |

Tricks & Shortcuts - Geometric Progressions –

Questions |

Formulas |

How to Solve Quickly |

Tricks & Shortcuts - Harmonic Progressions –

Questions |

Formulas |

How to Solve Quickly |

Tricks & Shortcuts

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