Arithmetic Progression MCQ Quiz - Objective Question with Answer for Arithmetic Progression - Download Free PDF

Given:

Series: 1/7, 1/11, 1/15, ...

Formula used:

This is an AP in the denominators: 7, 11, 15, ...

General term: T_n = 1 / [4n + 3]

Calculation:

9th term = 1 / (4×9 + 3) = 1 / 39

12th term = 1 / (4×12 + 3) = 1 / 51

Sum = 1/39 + 1/51

LCM of 39 and 51 = 663

1/39 = 17/663, 1/51 = 13/663

Sum = (17 + 13)/663 = 30/663

Simplify: 30 ÷ 3 = 10, 663 ÷ 3 = 221

∴ The sum is 10/221

Given:

Two-digit natural numbers divisible by 3

Formula used:

Use Arithmetic Progression (AP)

Sum = n/2 × (first term + last term)

Where: n = number of terms

Calculation:

Smallest 2-digit number divisible by 3 = 12

Largest 2-digit number divisible by 3 = 99

AP: 12, 15, 18, ..., 99

Common difference (d) = 3

To find number of terms (n):

Last term = a + (n - 1)d

99 = 12 + (n - 1) × 3

⇒ 99 - 12 = (n - 1) × 3

⇒ 87 = (n - 1) × 3

⇒ n - 1 = 29

⇒ n = 30

Sum = 30/2 × (12 + 99)

⇒ 15 × 111 = 1665

∴ The sum is 1665

Given:

Arithmetic Sequence 13, 19, 25, ...

Formula used:

Sum of first n terms of an Arithmetic Sequence: \(S_n = \dfrac{n}{2} \times (2a + (n-1)d)\)

Where:

S_n = Sum of n terms

n = Number of terms

a = First term

d = Common difference

Calculation:

The sequence 13, 19, 25, ...:

a = 13, d = 6, n = 30

S₃₀ = \(\dfrac{30}{2} \times (2 \times 13 + (30-1) \times 6)\)

⇒ S₃₀ = \(\dfrac{30}{2} \times (26 + 174)\)

⇒ S₃₀ = \(\dfrac{30}{2} \times 200\)

⇒ S₃₀ = 15 × 200

⇒ S₃₀ = 3000

∴ The correct answer is option (1).

Given:

The nth term of a sequence with first term a and common difference d is:

a_n = a + (n − 1)d

The sum of n terms of an AP is:

S_n = (n/2) × [2a + (n − 1)d]

Formula used:

Sum of first 3n terms: S_3n = (3n/2) × [2a + (3n − 1)d]

Sum of next n terms = S_n = S_4n − S_3n

Given: S_3n = S_n ⇒ 2S_3n = S_4n

Calculations:

Substitute sum formulas:

2 × (3n/2) × [2a + (3n − 1)d] = (4n/2) × [2a + (4n − 1)d]

Cancel 2:

3n × [2a + (3n − 1)d] = 2n × [2a + (4n − 1)d]

Expand both sides:

3n(2a + 3nd − d) = 2n(2a + 4nd − d)

6a + 9nd − 3d = 4a + 8nd − 2d

Now simplify:

6a − 4a = 8nd − 9nd + (−2d + 3d)

2a = d(1 − n) ......... (i)

Find the ratio of sum of first 2n terms to next 2n terms:

Required Ratio = S_2n / (S_4n − S_2n)

S_2n = (2n/2) × [2a + (2n − 1)d] = n × [2a + (2n − 1)d]

S_4n = (4n/2) × [2a + (4n − 1)d] = 2n × [2a + (4n − 1)d]

So, S_4n − S_2n = 2n[2a + (4n − 1)d] − n[2a + (2n − 1)d]

= n[4a + 8nd − 2d − 2a − 2nd + d]

= n[2a + 6nd − d]

Now substitute 2a from equation (i):

2a = (1 − n)d

S_2n = n[(1 − n)d + (2n − 1)d] = n × nd

S_4n − S_2n = n × 5nd

Final Ratio:

S_2n / (S_4n − S_2n) = nd / 5nd = 1 : 5

∴ The correct answer is Option 2: 1 : 5

Given:

Sequence: 3, 8, 13, 18, ...

Find the term where value = 78

Formula used:

nth term of an arithmetic sequence: a_n = a + (n-1)d

Where, a = first term = 3 and d = common difference = 8 - 3 = 5

Calculation:

78 = 3 + (n - 1) x 5

⇒ 78 - 3 = (n - 1) x 5

⇒ 75 = (n - 1) x 5

⇒ n - 1 = 75 ÷ 5

⇒ n - 1 = 15

⇒ n = 15 + 1

⇒ n = 16

∴ The correct answer is option (3).

Given:

13 + 23 + …….. + 93 = ?

Formula:

S_n = n/2 [a + l]

T_n = a + (n – 1)d

n = number of term

a = first term

d = common difference

l = last term

Calculation:

a = 13

d = 23 – 13 = 10

T_n = [a + (n – 1)d]

⇒ 93 = 13 + (n – 1) × 10

⇒ (n – 1) × 10 = 93 – 13

⇒ (n – 1) = 80/10

⇒ n = 8 + 1

⇒ n = 9

S₉ = 9/2 × [13 + 93]

= 9/2 × 106

= 9 × 53

= 477

Formula used:

an = a + (n – 1)d

Here, a → first term, n → Total number, d → common difference, an → n^th term

Calculation:

First three-digit number divisible by 6, (a) = 102 

Last three-digit number divisible by 6, (an) = 996

Common difference, (d) = 6 (Since the numbers are divisible by 6)

Now, a_n = a + (n – 1)d

⇒ 996 = 102 + (n – 1) × 6 

⇒ 996 – 102 = (n – 1) × 6

⇒ 894 = (n – 1) × 6

⇒ 149 = (n – 1)

⇒ n = 150

∴ The total three digit number divisible by 6 is 150

Given:

First term 'a' = 5, common difference 'd' = 4

Number of terms 'n' = 20

Concept:

Arithmetic progression:

• Arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.
• The fixed number is called common difference 'd'.
• It can be positive, negative or zero.

Formula used:

n^th term of AP 

T_n = a + (n - 1)d

The Sum of n terms of AP is given by

\(S = \dfrac{n}{2}[2a + (n-1)d]\)

\(S = \dfrac{n}{2}( a + l)\)

Where, 

a = first term of AP, d = common difference, l = last term 

Calculation:

We know that sum of n terms of AP is given by

\(S = \dfrac{n}{2}[2a + (n-1)d]\)

\(⇒ S = \dfrac{20}{2}[2× 5 + (20-1)× 4]\)

⇒ S = 10(10 + 76)

⇒ S = 860

Hence, the sum of 20 terms given AP will be 860.

We know that n^th term of AP is given by

Tn = a + (n - 1)d

If l is the 20th term (last term) of AP, then

l = 5 + (20 - 1) × 4 = 81

So the sum of AP

\(S = \dfrac{n}{2}( a + l)\)

\(⇒ S = \dfrac{20}{2}(5 + 81)\)

⇒ S = 860

Given condition: 

Numbers between 300 and 1000 are divisible by 7.

Concept:

Arithmetic Progression

a_n = a + (n - 1)d 

Calculation:

The first number that is divisible by 7 (300 - 1000) = 301 

Likewise: 301, 308, 315, 322...........994 

The above series makes an AP,

Where a = 301, Common Difference/d = 308 - 301 = 7 and Last term (an) = 994

⇒ an = a + (n - 1)d

⇒ 994 = 301 + (n - 1)7 

⇒ (994 - 301)/7 = n - 1 

⇒ 693/7 + 1 = n 

⇒ 99 + 1 = n 

⇒ n = 100 

∴ There are 100 numbers between 300 and 1000 which are divisible by 7. 

Given

2, 7, 12, ____________

Concept used

Tn = a + (n - 1)d

Where a = first term, n = number of terms and d = difference

Calculation

in the given series

a = 2

d = 7 - 2 = 5

T₁₀ = 2 + (10 - 1) 5

T₁₀ = 2 + 45

T₁₀ = 47

Tenth term = 47

Given: 

For a value of k; 2, 3 + k and 6 are to be in A.P 

Concept:

According to Arithmetic progression, a2 - a1 = a3 - a2 

where a1 ,a2 ,a3 are 1^st, 2^nd and 3^rd term of any A.P.

Calculation:

a₁ = 2, a₂ = k + 3, a₃ = 6 are three consecutive terms of an A.P.

According to Arithmetic progression, a₂ - a₁ = a₃ - a₂ 

(k + 3) – 2 = 6 – (k + 3)

⇒ k + 3 - 8 + k + 3 = 0

⇒ 2k = 2

After solving, we get k = 1

Given:

An AP is given
3 + 7 + 11 + 15 + 19 + ... upto 80 terms 

Formula used:

Sum of n^th term of an AP

S_n = (n/2){2a + (n - 1)d}

where, 

'n' is Number of terms, 'a' is First term, 'd' is Common difference

Calculations:

According to the question, we have

Sn = (n/2){2a + (n - 1)d}      ----(1) 

where, a = 3, n = 80, d = 7 - 3 = 4

Put these values in (1), we get

⇒ S80 = (80/2){2 × 3 + (80 - 1) × 4}

⇒ S80 = 40(6 + 79 × 4)

⇒ S80 = 40 × 322

⇒ S80 = 12,880

∴ The sum of 80^th terms of an AP is 12,880.

Alternate Method

n^th term = a + (n - 1)d

Here n = 80, a = 3 and d = 4

⇒ 80th term = 3 + (80 - 1)4

⇒ 80th term = 3 + 316

⇒ 80th term = 319

Now, the sum of n^th terms of an AP

⇒ S_n = (n/2) × (1st term + Last term)

⇒ S80 = (80/2) × (3 + 319)

⇒ S80 = 40 × 322

⇒ S80 = 12,880

∴ The sum of 80th terms of an AP is 12,880.

Concept used:

Let a, b, c… and so on be our series

As we know the common difference = b – a, c – b.

The common difference is the same in arithmetic progression

b – a = c – b

Calculation:

b – a = c – b

⇒ b + b = c + a

⇒ 2b = c + a

⇒ 2b = a + c

∴ a, b, c are in arithmetic progression then 2b = a + c.

Alternate Method

Let number be 1, 2, 3 which are in AP

 Only one option satisfied the equation 

2(2 ) 1 + 3 =so 2b = a + c is correct option

Concept used:

A sequence of numbers is called an arithmetic progression if the difference between any two consecutive terms is always same. 

Calculation:

In option 1, we have

Difference between 1^st and 2^nd term = 1 - 1 = 0

And, difference between 2nd and 3^rd term = 2 - 1 = 1

Here, the difference between any two consecutive terms is not equal.

So, option 1 is not in AP.

In option 2, we have

Difference between 1st and 2nd term = 0.33 - 0.3 = 0.03

And, difference between 2nd and 3rd term = 0.333 - 0.33 = 0.003

Here, the difference between any two consecutive terms is not equal.

So, option 2 is not in AP.

Now, in option 3, we have

Difference between 1st and 2nd term = 2 - √2 

⇒ 2 - 1.414

⇒ 0.586

And, difference between 2nd and 3rd term = 2√2 - 2

⇒ 2(1.414 - 1)

⇒ 2 × 0.414

⇒ 0.825

Here, the difference between any two consecutive terms is not equal.

So, option 3 is not in AP.

In option 4, we have

Difference between 1st and 2nd term = (3 + √2) - 3

⇒ √2

And, difference between 2nd and 3rd term = (3 + 2√2) - (3 + √2) 

⇒ √2

And, difference between 2nd and 3rd term = (3 + 2√2) - (3 + √2) 

⇒ √2

And, difference between 3nd and 4^th term = (3 + 3√2) - (3 + 2√2) 

⇒ √2

∴ Option 4 is in AP.

Mistake Points

In this question go through all the option and check with every term. Don't check for only first three options, 

Given :

A sequence 255, 264, 273, ...

Formula used :

Sum in A.P, Sn = \(\dfrac{n}{2}\times [2a+(n-1)d]\) 

Calculation :

n = 155 , a = 255 , d = 264 - 255 = 9

Sn = \(\dfrac{n}{2}\times [2a+(n-1)d]\)

Sn = \(\dfrac{155}{2}\times [2\times 255 + (155 - 1)9]\)

⇒\(\dfrac{155}{2}\times [510+1386]\)

⇒ 155 x 948 = 146940

∴ The answer is 146940.