# How To Solve Arithmetic Progression Questions Quickly

## Solve Arithmetic Progression Questions Quickly

Although we know Arithmetic Progression but most of us don’t know How to Solve Arithmetic Progression Questions.This is very important to Know for us.

In this Page How to Solve Arithmetic Progression Questions is given along with their Solutions. This will very useful for students Preparing for different Examination.

### How to Solve Arithmetic Progression questions easily

An AP is represented in the form a, (a + d), (a + 2d), (a + 3d), …
where a = the first term, and  d = the common difference.

n is number of Terms

General form, Tn = a + (n-1)d

Where Tn is nth term of an Arithmetic Progresion

### Type 1: Find nth term of an AP: tn = a + (n – 1)d

Question 1. If tn = 20, d = -1 and n = 16, then the first term is?

Options:

A. 36

B.-35

C. 35

D. 5

Solution:    We know that,  tn = a + (n – 1)d

20 = a + (16 – 1) (-1)

20 = a + (15) (-1)

20 = a – 15

-a = – 15 – 20

– a = – 35

a = 35

Correct option: C

Question 2. What is the position of 62 in the following series 2, 5, 8, ….?

Options:

A. 21

B. 22

C. 23

D. 20

Solution:     In the given series,

a = 2

d = 3

tn = 62

tn = a + (n – 1)d

62 = 2 + (n -1) 3

62 = 2 + 3n – 3

62 – 2 + 3 = 3n

63 = 3n

n = 21

Correct option: A

Question 3. Find the 11th term of the series 1, 3.5, 6, 8.5, ……

Options:

A. 25

B. 26

C. 23.5

D. 24.5

Solution:    In the given series a = 1, d = 2.5

n =11

tn = a + (n – 1)d

tn = 1 + (11 – 1) 2.5

tn = 1 + 10 x 2.5

tn = 1+ 25

tn = 26

Correct option: B

### Type 2: Number of terms in the series: $n = \left [ \frac{l – a}{d} \right ] + 1$

Question 1.Find the number of terms in the series 2, 4, 6, . . .98

Options:

A. 50

B. 49

C.51

D. 48

Solution     We know that, $n = \left [ \frac{l – a}{d} \right ] + 1$

a (first term) = 2

l (last term) = 98

d (common difference) = 2

$n= [\frac{(98-2)}{2} ]+ 1$

n = 48 + 1

n = 49

Correct option: B

Question 2.Find the last number of the series where, a = 3, d = 5, and number of terms are 45

Options:

A. 225

B.223

C.230

D. 222

Solution     According to the question,

a = 3, d = 5, n = 45

$n = \left [ \frac{l – a}{d} \right ] + 1$

45 = $[\frac{(l-3)}{5}]+ 1$

$45-1 = \frac{(l-3)}{5}$

44 x 5 = l – 3

220 = l – 3

l = 223

Correct option: B

Question 3.Find the total odd numbers  in the series 3….. 99.

Options:

A. 49

B.51

C. 50

D. 60

Solution:    The common difference is 2 because it is an odd number series.

We know that,

$n = \left [ \frac{l – a}{d} \right ] + 1$

a = 3

l = 99

d (common difference) = 2

$n = \frac{(99-3)}{2} + 1$

n = $\frac{96}{2} + 1$

n = 48 + 1

n = 49

Correct option: A

### Type 3: Sum of first ‘n’ terms of the series: $S_{n} = \frac{n}{2}\left [ 2a + (n-1) d\right ] \text{ or } \frac{n}{2}(a + l)$

Question 1. Sanjana has one paisa on 1st of January with her. She got 2 paisa on the 2nd January and 3 paisa on 3rd of January. In similar fashion, what will be her saving after 365 days?

Options:

A. 66700 paisa

B. 66795 paisa

C. 65795 paisa

D. 66790 paisa

Solution:     According to the question, the series formed is 1,2,3,4,5,……

Therefore, a = 1 and d = 1

$S_{n} = \frac{n}{2}\left [ 2a + (n-1) d\right ]$

n = 365

S365 $\frac{365}{2}\times [2 \times 1 + (365 − 1) \times 1]$

S365 = 66795 paisa

Correct option: C

Question 2.Find the sum of the series 62 + 60 + 58 …….. + 30

Options:

A. 728

B.780

C.719

D. 782

Solution:    In the given series a = 62,  d = -2,  tn = 30

tn = a + (n – 1)d

30= 62 + (n – 1)-2

30 = 62 – 2n + 2

2n = 62-30+2

34 = 2n

n = 17

$S_{n} = \frac{n}{2}(a + l)$

S17 = $\frac{17}{2}\times (62 + 30)$

S17 $\frac{17}{2}\times (92)$

S17 = 782

Correct option: D

Question 3.If the sum of 5 consecutive numbers is 100 then find out the first number.

Options:

A. 20

B. 25

C. 18

D. 16

Solution:     It is clear from the question that d = 1 because there are 5 consecutive numbers

n = 5

Sn= 100

Let the first number ‘a’ be x

$S_{n} = \frac{n}{2}\left [ 2a + (n-1) d\right ]$

100 = $\frac{5}{2} [2a + (5 − 1) \times 1]$

$100 = \frac{5}{2}[2a + (4) \times 1]$

$100\times \frac{2}{5} = [2a + 4]$

40 = 2a + 4

a = 18

So, the first number is 18 and the complete series is 18, 19, 20, 21, and 22.

Correct option: C

### Type 4: Arithmetic mean of the series: $b = \frac{(a + c)}{2}$

Question 1. Find the Arithmetic mean of 10 and 18.

Options:

A. 14

B. 15

C. 16

D. 17

Solution:    We know that,  Arithmetic mean of a and b is $\frac{a+b}{2}$

Therefore, there arithmetic mean (AM) = $\frac{10+18}{2} = 14$

Correct option: A

Question 2. Find the arithmetic mean of first 11 whole numbers.

Options:

A. 4

B. 6

C.5

D.6

Solution:    Here First term = 0

Last term = 10

Therefore, arithmetic mean (AM) = $\frac{0+1+2+3+4+5+6+7+8+9+10}{11} = 5$

Correct option: C

Question 3. Find the arithmetic mean of series 2, 4, 6 , 8….100

Options:

A. 51

B.50

C.25

D.49

Solution:     In the given series, a = 2, d = 2, l = 100

To find the total number of term we can use the formula

$n= [\frac{(l-a)}{d}] + 1$

n =  $[\frac{(100-2)}{2}]+ 1$

n = 50

Now, to find the sum of the series we can use the formula Sn =$\frac{n}{2} [2a + (n − 1) d]$  OR  $\frac{n}{2}(a+l)$

Sn = $\frac{n}{2} (a+l)$

Sn $\frac{n}{2}(a+l)$

S50$\frac{50}{2}(2+l00)$

S50 = 25 (102)

S50 = 2550

Now, AM = $\frac{2550}{50} = 51$

Correct option: A

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