Arithmetic Progressions Shortcut, tricks, and tips

September 9, 2023

Arithmetic Progression tricks, shortcuts, and tips

Arithmetic Progression can applied in real life by analyzing a certain pattern that we see in our daily life. In this page tips and tricks for Arithmetic Progression as well as different types of questions and their answers is given for solving questions of arithmetic progression in less time.

Arithmetic Progression tricks If the initial term of an arithmetic progression is a, and the common difference of successive members is d, then the n-th term of the sequence is given by

a_{n}

= a + (n − 1)d, n = 1, 2,..and so on

Tips and Tricks for Arithmetic Progression To Solve Arithmetic Progression in Aptitude

An Arithmetic Progression or Arithmetic Sequence is a sequence of numbers or terms in a way that the difference between the consecutive terms is constant. Here, are some easy tips and tricks for you solve Arithmetic Progression questions quickly, easily, and efficiently in competitive exams.

An Arithmetic Progression 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, T_n = a + (n-1)d

Where T_nis nth term of an Arithmetic Progression

There are 4 types of questions asked in exams.

Properties of Arithmetic Progression

If a fixed number is added or subtracted from each term of an AP, then the resulting sequence is also an AP and it has the same common difference as that of the original AP.
If each term in an AP is divided or multiply with a constant non-zero number, then the resulting sequence is also in an AP.
If n^thterm is in linear expression then the sequence is in AP.
If a₁, a₂, a₃, …, a_nand b₁, b₂, b₃, …, b_n, are in AP. then a₁+b₁, a₂+b₂, a₃+b₃, ……, a_n+b_nand a₁–b₁, a₂–b₂, a₃–b₃, ……, a_n–b_n will also be in AP.
If n^thterm of a series is T_n = An + B, then the series is in AP
Three terms of the A.P whose sum or product is given should be assumed as a-d, a, a+d.
Four terms of the A.P. whose sum or product is given should be assumed as a-3d, a-d, a+d, a+3d.

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Type 1: Find nth term of series $t_{n} = a+(n-1)d$

Question 1 : Find 10^th term in the series 1, 3, 5, 7, …

                      Options
                      A 20
                      B 19
                      C 15
                      D 21

Solution: We know that,

t_n = a + (n – 1)d
where t_n = nth term,

a= the first term ,

d= common difference,

n = number of terms in the sequence

In the given series,

a (first term) = 1

d (common difference) = 2 (3 – 1, 5 – 3)

Therefore, 10^th term = t₁₀ = a + (n-1) d

t₁₀ = 1 + (10 – 1) 2

t₁₀ = 1 + 18

t₁₀ = 19

Correct option: B

Question 2 : Find last term in the series if there are 8 term in this series 13 , 17 , 21 ,25….

                      Options
                      A 33
                      B 41
                      C 37
                      D 39

Solution: We know that,

t_n = a + (n – 1)d
where t_n = nth term,

a = the first term ,

d = common difference,

n = number of terms in the sequence

In the given series,

a (first term) = 13

d (common difference) = 4(17 – 13, 21 – 17)

Therefore, 8^th term = t₈= a + (n-1) d

t₈ = 13 + (8 – 1) 4

t₈ = 13 + 28

t₈ = 41

Correct option: B

Type 2: Find number of terms in the series $n = \frac{l-a}{d} +1$

Question 1 : Find the number of terms in the series 7, 11, 15, . . .71

Options
A 12
B 25
C 22
D 17

Solution: We know that, $n= [ \frac{(l-a)}{d} ]+ 1$

where n = number of terms,

a= the first term,

l = last term,

d= common difference

In the given series,

a (first term) = 7

l (last term) = 71

d (common difference) = 11 – 7 = 4

n= [ $\frac{(71-7)}{4}]+ 1$

n = $\frac{64}{4} + 1$

n = 16 + 1

n = 17

Correct option: D

Question 2 : Find the number of terms if First term = 22 ,Last term = 50 and common difference is 4

Options:

                       A 10
                       B 9
                       C 8
                       D 7

Solution: We know that,

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

where n = number of terms,

a = the first term,

l = last term,

d = common difference

In the given series,

a (first term) = 22

l (last term) = 50

d (common difference) = 4

n = $[\frac{(50-22)}{4}]+ 1$

n = $\frac{28}{4} +1$

n = 7+ 1

n = 8

Correct option: C

Type 3: Find sum of first ‘n’ terms of the series $S_{n} =\frac{n}{2}[2a+(n-1)\times d]$ or $\frac{n}{2}(a+l)$

Question 1. Find the sum of the series 1, 3, 5, 7…. 201

                   Options
                   A 12101
                   B 25201
                   C 22101
                   D 10201

Solution: We know that,

S_n = $\frac{n}{2}[2a + (n − 1) d]$

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

a = the first term,

d= common difference,

l = t_n = n^th term = a + (n-1)d

In the given series,

a = 1, d = 2, and l = 201

Since we know that, l = a + (n – 1) d

201 = 1 + (n – 1) 2

201 = 1 + 2n -2

202 = 2n

n = 101

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

S_n = $\frac{101}{2}$ (1+201)

S_n = 50.5 (1 + 201)

S_n = 50.5 x 202

S_n = 10201

Correct option: D

Question 2. Find the sum of the Arithmetic series if First term of this series is 45 , common difference is 5 and number of terms in this series is 8.                   Options
                   A 500
                   B 300
                   C 400
                   D 200

Solution: We know that,

S_n = $\frac{n}{2}[2a + (n − 1) d]$

$\frac{n}{2} (a+l)$
where, a = the first term,

d = common difference,

l = t_n = n^th term = a + (n-1)d

In the given series,

a = 45, d = 5, and n = 8

S_n= $\frac{n}{2}[2a + (n − 1) d]$

S_n = $\frac{8}{2}[2\times 45 + (8 − 1) \times 5]$

S_n = 4 (90 + 35)

S_n = 4 x 125

S_n = 500

Correct option: A

Type 4: Find the arithmetic mean of the series. $b = \frac{a+c}{2}$

Question 1.Find the arithmetic mean of first five prime numbers.

Options:

                   A 6.6
                   B 3.6
                   C 5.6
                   D 7.6

Solution: We know that

b = $\frac{1}{2} (a + c)$

Here, five prime numbers are 2, 3, 5, 7 and 11

Therefore, their arithmetic mean (AM) = $\frac{(2+3+5+7+11)}{5} = 5.6$

Correct option: C

Question 2.Find Second Number if arithmetic mean of two numbers is 24 and First number is 10.

Options:

                   A 32
                   B 38
                   C 34
                   D 36

Solution: We know that

b = $\frac{1}{2} (a + c)$

Let second Number is b

Therefore, their arithmetic mean (AM) = $\frac{(10 + b)}{2} = 24$

b = 38

Correct option: B

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2 comments on “Arithmetic Progressions Shortcut, tricks, and tips”

Sunishtha
plz check your type 3 question 1 answer and option don’t match

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HelpPrepInsta
Really sorry for the silly mistake Sunishtha, we’ll fix it up

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Arithmetic Progressions Shortcut, tricks, and tips

Arithmetic Progression tricks, shortcuts, and tips