# How To Solve Arthmetic And Harmonic Progression Quickly

**Definition and Solve Quickly**

**AP**

An arithmetic progression is a sequence of numbers in which each term is derived from the preceding term by adding or subtracting a fixed number called the common difference “d”.

**GP**

A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio.

**HP**

A series of terms is known as a HP series when their reciprocals are in arithmetic progression.

**Read Also**–**Formulas to solve AP GP & HP questions**

**Type 1: ****AP questions**

### Question 1.

**Find the first term of the AP series in which 10 ^{th} term is 6 and 18^{th} term is 70. **

Options:

- 76
- – 76
- 66
- – 66

#### Solution:

10^{th} term = (a + 9d) = 6….(1)

18^{th} term = (a + 17d) = 70 ……. (2)

On solving equation 1 and 2

We get, d = 8

Put the value of d in equation 1

(a + 9d) = 6

a + 9 * 8 = 6

a + 72 = 6

a = -66

#### Correct option: D

### Question 2.

**Find the n ^{th} term of the series 3, 8, 13, 18,…,**

Options:

- 2(2n+ 1)
- 5n + 2
- 5n – 2
- 2(2n – 1)

#### Solution:

The given series is in the form of AP.

first term a = 3

common difference d = 5

We know that, n^{th} term = t_{n} = a + (n-1)d

Therefore, t_{n} = 3 + (n-1) * 5

= 3 + 5n – 5

= 5n – 2

#### Correct option: C

### Question 3.

**The series 28, 25,……. -29 has 20 terms. Find out the sum of all 20 terms?**

Options:

- -10
- -12
- 10
- 12

#### Solution:

a =28, d= -3 (25 – 28), l = -29, n = 20

Sum of all n-terms = S_{n} = n (a+l)/2

S_{20} = 20 (28 + (-29)) / 2

S_{20} = -10

#### Correct option: A

**Type 2: GP questions**

### Question 1.

**Find the sum of the following infinite G. P. 1/3, 1/9, 1/27, 1/81…….**

Options:

- 1/3
- 2/3
- 1/5
- 1/2

#### Solution:

a = 3, r = 1/9/1/3 = 1/3

Required sum = a/(1-r)

= 1/3 / (1-1/3)

= 1/3 / 2/3

= ½

#### Correct option: D

### Question 2.

**Find the G. M. between 4/25 and 196/25**

Options:

- 28/5
- 28/25
- 8/25
- 14/5

#### Solution:

Geometric mean √ab

GM = √4/25 * √196/25

GM = 2/5 * 14/5

GM = 28/25

#### Correct option: B

### Question 3.

**Find the number of terms in the series 1, 3, 9 , ….19683**

Options:

- 10
- 8
- 6
- 7

#### Solution:

In the given series,

a_{1} = 1, r = 3/1 = 3, a_{n }=19683

=

19683 = 1* (3^{n-1})

19683 = 3^{n-1}

3^{9} = 3^{n-1}

9 = n-1

n = 10

#### Correct option: A

**Type 3: HP questions**

### Question 1:

**If the 6 ^{th} term of H.P. is 10 and the 11^{th} term is 18. Find the 16^{th} term.**

Options:

- 90
- 110
- 85
- 100

#### Solution:

6^{th} term = a + 5d = 1/10……(1)

11^{th} term = a + 10d = 1/18……(2)

On solving equation 1 and 2 we get, d = -2/225

Put value of d in equation 1

a + 5d = 1/10

a + 5 * -2/225 = 1/10

a = 13/90

Now, 16^{th} term = a + 15d = 13/ 90 + 15 * – 2/55

= 13/90 – 30/225

= 1/90

Therefore 16^{th} term = 90

#### Correct option: A

### Question 2.

**Find the Harmonic mean of 6, 12, 18**

Options:

- 10.12
- 9.62
- 9.81
- 8.10

#### Solution:

We know that,

HM=n/s

where , s=(1/a)+(1/b)+(1/c)

s=(1/6)+(1/12)+(1/18)

=(11/36)

HM = n/s=3/(11/36)

HM = 3*36/11

HM=108/11

HM=9.81

#### Correct option: C

### Question 3.

**What is the relation between AM, GM, and HM?**

Options:

- AM * HM = GM
^{2} - AM / HM = GM
- AM + HM = GM
^{2} - AM – HM = GM
^{2}

#### Solution:

AM = a+b/2

GM = √ab

HM = 2ab/a+b

Therefore AM * HM = GM^{2}

a+b/2 * 2ab/a+b = ab

Let P = {2, 3, 4, ………. 100} and Q = {101, 102, 103, ….. 200}. How many elements of Q are there such that they do not have any element of P as a factor ?

what is the method to find the prime no. between the numbers as soon as possible ?

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type 3 harmonic progression. Please explain

Question 2.

Find the Harmonic mean of 6, 12, 18

Options:

10.12

10.9

10.06

6.10

Solution:

We know that,

HM = 3/0.298

HM = 10.06

Correct option: C

formula= n/(1/a1 +1/a2 + 1/a3)

6,12,18 we have 3 numbers

so n=3

then 3/(1/6 + 1/12 +1/18)

ormula= n/(1/a1 +1/a2 + 1/a3)

6,12,18 we have 3 numbers

so n=3

then 3/(1/6 + 1/12 +1/18)

after using this formula we get 9.868 answer

Find the Harmonic mean of 6, 12, 18

Options:

10.12

10.9

10.06

6.10

Solution:

We know that,

HM = 3/0.298

HM = 10.06 ——-???

Please explain

type 3 harmonic progression.

Question 2.

Find the Harmonic mean of 6, 12, 18 Options: 10.12

10.9

10.06

6.10

Solution:

We know that, HM = 3/0.298 HM = 10.06 Correct option: C

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