Question 1

What is the next number in the series: 1, 4, 9, 16, 25, ?

Accepted Answer

Step 1: Recognize that each term is a perfect square: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25. Step 2: The series consists of consecutive perfect squares. Step 3: The next term should be 6² = 36. Therefore, the answer is 36.

Question 2

What is the next term in the series: 2, 6, 12, 20, 30, ?

Accepted Answer

Step 1: Find the differences between consecutive terms: 6−2=4, 12−6=6, 20−12=8, 30−20=10. Step 2: The first differences form the series 4, 6, 8, 10 (increasing by 2 each time). Step 3: The next first difference should be 12. Step 4: 30 + 12 = 42. Therefore, the next term is 42.

Question 3

Find the missing number: 3, 9, 27, 81, 243, ?

Accepted Answer

Step 1: Observe the relationship between consecutive terms. Step 2: 9÷3 = 3, 27÷9 = 3, 81÷27 = 3, 243÷81 = 3. Step 3: Each term is multiplied by 3 to get the next term (geometric progression with r = 3). Step 4: 243 × 3 = 729. Therefore, the missing number is 729.

Question 4

Find the missing number in the series: 5, 10, 20, 40, 80, ?

Accepted Answer

Step 1: Identify the pattern by examining the ratio between consecutive terms. Step 2: 10÷5 = 2, 20÷10 = 2, 40÷20 = 2, 80÷40 = 2. Step 3: Each term is multiplied by 2 to get the next term. Step 4: 80 × 2 = 160. Therefore, the missing number is 160.

Question 5

What is the missing term in the series: 100, 90, 81, 73, 66, ?

Accepted Answer

Step 1: Calculate the differences between consecutive terms: 100−90=10, 90−81=9, 81−73=8, 73−66=7. Step 2: The differences form the sequence 10, 9, 8, 7 (decreasing by 1 each time). Step 3: The next difference should be 6. Step 4: 66 − 6 = 60. Therefore, the missing term is 60.

Question 6

Find the next term in the series: 2, 5, 10, 17, 26, ?

Accepted Answer

Step 1: Find the differences between consecutive terms: 5-2=3, 10-5=5, 17-10=7, 26-17=9. Step 2: The first differences are 3, 5, 7, 9 (odd numbers increasing by 2). Step 3: The next first difference should be 11. Step 4: Therefore, the next term = 26 + 11 = 37. Therefore, the answer is 37.

Question 7

What is the missing number in the series: 5, 10, 20, 40, ?, 160?

Accepted Answer

Step 1: Observe the pattern between consecutive terms. Step 2: 10÷5 = 2, 20÷10 = 2, 40÷20 = 2. Step 3: This is a geometric series with common ratio 2. Step 4: The missing term = 40 × 2 = 80. Step 5: Verify: 80 × 2 = 160 ✓. Therefore, the answer is 80.

Question 8

Find the missing number in the series: 2, 6, 18, 54, ?

Accepted Answer

Step 1: Identify the pattern. Each term is multiplied by 3 to get the next term. Step 2: 2 × 3 = 6, 6 × 3 = 18, 18 × 3 = 54, 54 × 3 = 162. Therefore, the missing number is 162.

Question 9

Find the next term: 2, 3, 5, 8, 13, ?

Accepted Answer

Step 1: Examine the relationship between consecutive terms. Step 2: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13. Step 3: Each term is the sum of the two preceding terms (Fibonacci-like sequence). Step 4: 8 + 13 = 21. Therefore, the next term is 21.

Question 10

Find the next number in the series: 3, 6, 12, 24, 48, ?

Accepted Answer

Step 1: Identify the pattern by examining the ratio between consecutive terms. Step 2: 6÷3 = 2, 12÷6 = 2, 24÷12 = 2, 48÷24 = 2. Step 3: Each term is multiplied by 2 to get the next term. Step 4: 48 × 2 = 96. Therefore, the answer is 96.

Question 11

A series is: 5, 11, 23, 47, 95, ?. The next term is:

Accepted Answer

Step 1: Find the differences: 11−5=6, 23−11=12, 47−23=24, 95−47=48. Step 2: The differences are 6, 12, 24, 48 (each is double the previous). Step 3: The next difference = 48 × 2 = 96. Step 4: The next term = 95 + 96 = 191. Therefore, the answer is 191.

Question 12

In a number series, consecutive differences form a pattern: 2, 5, 10, 17, 26, ?. What is the next term?

Accepted Answer

Step 1: Find the differences between consecutive terms: 5−2=3, 10−5=5, 17−10=7, 26−17=9. Step 2: The differences are 3, 5, 7, 9 (odd numbers increasing by 2). Step 3: The next difference should be 11. Step 4: Next term = 26 + 11 = 37. Therefore, the answer is 37.

Question 13

In a series, each term is the sum of the previous two terms: 2, 3, 5, 8, 13, ?, 34. The missing term is:

Accepted Answer

Step 1: Verify the Fibonacci-like pattern: 2+3=5, 3+5=8, 5+8=13. Step 2: The next term = 8 + 13 = 21. Step 3: Verify: 13 + 21 = 34 ✓. Therefore, the missing term is 21.

Question 14

A series is defined as: 3, 6, 12, 24, ?, 96. The missing term is:

Accepted Answer

Step 1: Observe the ratio between consecutive terms. 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. Step 2: This is a geometric series with common ratio r = 2. Step 3: The missing term = 24 × 2 = 48. Step 4: Verify: 48 × 2 = 96 ✓. Therefore, the missing term is 48.

Question 15

A series follows the pattern: 1, 4, 9, 16, 25, ?, 49. The missing term is:

Accepted Answer

Step 1: Recognize that each term is a perfect square: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25. Step 2: The series is n² where n = 1, 2, 3, 4, 5, ?, 7. Step 3: The missing position is n = 6. Step 4: The missing term = 6² = 36. Therefore, the answer is 36.

Question 16

A number series follows the pattern: 3, 6, 12, 24, ?, 96. What is the missing term?

Accepted Answer

Step 1: Observe the ratio between consecutive terms. 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. Step 2: This is a geometric series with common ratio r = 2. Step 3: The missing term = 24 × 2 = 48. Step 4: Verify: 48 × 2 = 96 ✓. Therefore, the missing term is 48.

Question 17

In a series, the difference between consecutive terms increases by 1 each time. If the first term is 2 and the second term is 3, what is the 6th term?

Accepted Answer

Step 1: First term a₁ = 2, second term a₂ = 3. Difference d₁ = 3 − 2 = 1. Step 2: The difference increases by 1 each time, so d₂ = 2, d₃ = 3, d₄ = 4, d₅ = 5. Step 3: a₃ = a₂ + d₂ = 3 + 2 = 5. Step 4: a₄ = a₃ + d₃ = 5 + 3 = 8. Step 5: a₅ = a₄ + d₄ = 8 + 4 = 12. Step 6: a₆ = a₅ + d₅ = 12 + 5 = 17. Therefore, the 6th term is 17.

Question 18

A series is defined as: a₁ = 10, and aₙ = aₙ₋₁ + n for n ≥ 2. What is the value of a₄?

Accepted Answer

Step 1: a₁ = 10 (given). Step 2: a₂ = a₁ + 2 = 10 + 2 = 12. Step 3: a₃ = a₂ + 3 = 12 + 3 = 15. Step 4: a₄ = a₃ + 4 = 15 + 4 = 19. Therefore, a₄ = 19.

Question 19

A series follows the pattern where T(n) = (n³ - n) / 2. However, if T(n) is even, it is divided by 2 in the final series. What is the value of T(5) in this modified series?

Accepted Answer

Step 1: Calculate T(5) using the formula T(n) = (n³ - n) / 2. Step 2: T(5) = (125 - 5) / 2 = 120 / 2 = 60. Step 3: Check if 60 is even. Yes, 60 is even. Step 4: Apply the modification: divide by 2. Step 5: Modified T(5) = 60 / 2 = 30. Therefore, the answer is 30.

Question 20

A number series has the property that T(n) = n² + 2n. However, every third term (T(3), T(6), T(9), ...) is replaced by the sum of its two neighbouring terms. What is T(6) in this modified series?

Accepted Answer

Step 1: Calculate original terms using T(n) = n² + 2n. T(1) = 1 + 2 = 3, T(2) = 4 + 4 = 8, T(3) = 9 + 6 = 15, T(4) = 16 + 8 = 24, T(5) = 25 + 10 = 35, T(6) = 36 + 12 = 48. Step 2: T(3) is a multiple of 3, so it is replaced: T(3) = T(2) + T(4) = 8 + 24 = 32. Step 3: T(6) is also a multiple of 3, so it is replaced: T(6) = T(5) + T(7). Step 4: Calculate T(7) = 49 + 14 = 63. Step 5: T(6) = 35 + 63 = 98. Therefore, the modified T(6) is 98.

Question 21

In a series, T(n) = T(n-1) × 2 - T(n-2) for n ≥ 3. Given T(1) = 4 and T(2) = 6, what is T(6)?

Accepted Answer

Step 1: Apply the recurrence T(n) = 2·T(n-1) - T(n-2). Step 2: T(3) = 2(6) - 4 = 12 - 4 = 8. Step 3: T(4) = 2(8) - 6 = 16 - 6 = 10. Step 4: T(5) = 2(10) - 8 = 20 - 8 = 12. Step 5: T(6) = 2(12) - 10 = 24 - 10 = 14. Therefore, T(6) = 14.

Question 22

A series is defined as: T(n) = T(n-1) + T(n-2) for n ≥ 3, with T(1) = 2 and T(2) = 3. What is the sum of the 4th and 5th terms?

Accepted Answer

Step 1: This is a Fibonacci-type recurrence. T(1) = 2, T(2) = 3. Step 2: T(3) = T(2) + T(1) = 3 + 2 = 5. Step 3: T(4) = T(3) + T(2) = 5 + 3 = 8. Step 4: T(5) = T(4) + T(3) = 8 + 5 = 13. Step 5: Sum = T(4) + T(5) = 8 + 13 = 21. Therefore, the answer is 21.

Question 23

In a number series, the first term is 2. Each subsequent term is obtained by multiplying the previous term by 3 and then subtracting 4. If the fifth term of this series is represented as T₅, find T₅.

Accepted Answer

Step 1: T₁ = 2. Step 2: T₂ = (2 × 3) − 4 = 6 − 4 = 2. Step 3: T₃ = (2 × 3) − 4 = 6 − 4 = 2. Step 4: T₄ = (2 × 3) − 4 = 6 − 4 = 2. Step 5: T₅ = (2 × 3) − 4 = 6 − 4 = 2. The series becomes constant at T₁ = 2, so T₅ = 2.

Question 24

In a number series, the difference between consecutive terms follows a pattern: the differences are 2, 6, 12, 20, ... If the first term is 5, what is the 6th term?

Accepted Answer

Step 1: Identify the difference pattern: 2, 6, 12, 20. Step 2: Find the second differences: 6-2=4, 12-6=6, 20-12=8. These increase by 2 each time. Step 3: The next differences are: 20+10=30, 30+12=42. Step 4: Build the series: T1=5, T2=5+2=7, T3=7+6=13, T4=13+12=25, T5=25+20=45, T6=45+30=75. Therefore, the 6th term is 75.

Question 25

Find the missing term in the series: 3, 7, 16, 35, 74, ?

Accepted Answer

Step 1: Identify the pattern by examining differences or relationships between consecutive terms. Step 2: Check if each term follows: next term = 2 × current term + 1. 3×2+1=7 ✓, 7×2+2=16 ✓ — let's re-examine. Step 3: Try: T(n) = 2×T(n-1) + T(n-2) pattern? 2×7+3=17 ✗. Step 4: Try differences: 7-3=4, 16-7=9, 35-16=19, 74-35=39. Differences: 4, 9, 19, 39 — each roughly doubles minus 1? 4×2+1=9 ✓, 9×2+1=19 ✓, 19×2+1=39 ✓, so next difference = 39×2+1=79. Step 5: Missing term = 74+79=153. Therefore, the answer is 153.

Question 26

A series is constructed where the nth term is defined as: T(n) = n² + 2n − 1. However, every third term (3rd, 6th, 9th, ...) is replaced by the sum of its two adjacent terms. What is the 9th term of the modified series?

Accepted Answer

Step 1: Calculate the original series using T(n) = n² + 2n − 1. T(1) = 1 + 2 − 1 = 2. T(2) = 4 + 4 − 1 = 7. T(3) = 9 + 6 − 1 = 14. T(4) = 16 + 8 − 1 = 23. T(5) = 25 + 10 − 1 = 34. T(6) = 36 + 12 − 1 = 47. T(7) = 49 + 14 − 1 = 62. T(8) = 64 + 16 − 1 = 79. T(9) = 81 + 18 − 1 = 98. Step 2: Apply the modification rule. Every third term is replaced by the sum of adjacent terms. Step 3: T(3) is replaced by T(2) + T(4) = 7 + 23 = 30. T(6) is replaced by T(5) + T(7) = 34 + 62 = 96. T(9) is replaced by T(8) + T(10). Step 4: Calculate T(10) = 100 + 20 − 1 = 119. Step 5: T(9) modified = T(8) + T(10) = 79 + 119 = 198.

Question 27

What is the missing number in the series: 5, 10, 20, 40, ?, 160?

Accepted Answer

Step 1: Examine the ratio between consecutive terms. 10÷5=2, 20÷10=2, 40÷20=2. Step 2: This is a geometric sequence with common ratio r=2. Step 3: Missing term = 40×2=80. Step 4: Verify: 80×2=160 ✓

Question 28

Find the next term in the series: 1, 4, 9, 16, 25, ?

Accepted Answer

Step 1: Recognize that each term is a perfect square. 1=1², 4=2², 9=3², 16=4², 25=5². Step 2: The series represents squares of consecutive natural numbers. Step 3: Next term = 6²=36.

Question 29

Find the next term in the series: 3, 6, 12, 24, 48, ?

Accepted Answer

Step 1: Identify the pattern by finding the ratio between consecutive terms. 6÷3 = 2, 12÷6 = 2, 24÷12 = 2, 48÷24 = 2. Step 2: This is a geometric series with common ratio 2. Step 3: Next term = 48 × 2 = 96.

Question 30

Find the missing number in the series: 2, 3, 5, 8, 13, ?, 34

Accepted Answer

Step 1: Check if each term is the sum of the two preceding terms. 2+3=5, 3+5=8, 5+8=13. Step 2: This is a Fibonacci-type sequence. Step 3: Missing term = 8+13=21. Step 4: Verify: 13+21=34 ✓

Question 31

What is the next number in the series: 3, 6, 11, 18, 27, ?

Accepted Answer

Step 1: Calculate first differences: 6−3=3, 11−6=5, 18−11=7, 27−18=9. Step 2: First differences are 3, 5, 7, 9 (odd numbers in arithmetic progression with common difference 2). Step 3: Next first difference = 9+2=11. Step 4: Next term = 27+11=38.

Question 32

Find the next number in the series: 2, 6, 12, 20, 30, ?

Accepted Answer

Step 1: Identify the pattern by finding differences. First differences: 6−2=4, 12−6=6, 20−12=8, 30−20=10. Step 2: The first differences form an arithmetic sequence (4, 6, 8, 10) with common difference 2. Step 3: Next first difference = 10+2=12. Step 4: Next term = 30+12=42.

Question 33

What is the next term in the series: 100, 50, 25, 12.5, ?

Accepted Answer

Step 1: Examine the ratio between consecutive terms. 50÷100=0.5, 25÷50=0.5, 12.5÷25=0.5. Step 2: This is a geometric sequence with common ratio r=0.5 (or dividing by 2 each time). Step 3: Next term = 12.5÷2=6.25.

Question 34

In a number series, the first term is 5 and each subsequent term is obtained by multiplying the previous term by 2 and then subtracting 3. What is the 5th term of this series?

Accepted Answer

Step 1: First term (a₁) = 5. Step 2: Second term (a₂) = 5 × 2 − 3 = 10 − 3 = 7. Step 3: Third term (a₃) = 7 × 2 − 3 = 14 − 3 = 11. Step 4: Fourth term (a₄) = 11 × 2 − 3 = 22 − 3 = 19. Step 5: Fifth term (a₅) = 19 × 2 − 3 = 38 − 3 = 35. Therefore, the 5th term is 35.

Question 35

Complete the series: 1, 4, 9, 16, 25, ?, 49

Accepted Answer

Step 1: Recognize that each term is a perfect square: 1=1², 4=2², 9=3², 16=4², 25=5². Step 2: The series consists of consecutive perfect squares. Step 3: Missing term = 6² = 36. Step 4: Verify: 7² = 49 ✓

Question 36

Find the next number in the series: 2, 5, 10, 17, 26, ?

Accepted Answer

Step 1: Find the differences between consecutive terms: 5-2=3, 10-5=5, 17-10=7, 26-17=9. Step 2: The differences form a series: 3, 5, 7, 9 (odd numbers increasing by 2). Step 3: Next difference = 9 + 2 = 11. Step 4: Next term = 26 + 11 = 37.

Question 37

In a series, the difference between consecutive terms increases by 1 each time. If the first term is 2 and the second term is 3, what is the 6th term?

Accepted Answer

Step 1: First term = 2, second term = 3. Difference = 3 − 2 = 1. Step 2: The differences form the sequence 1, 2, 3, 4, 5, ... Step 3: Term 3 = 3 + 2 = 5. Step 4: Term 4 = 5 + 3 = 8. Step 5: Term 5 = 8 + 4 = 12. Step 6: Term 6 = 12 + 5 = 17. Therefore, the 6th term is 17.

Question 38

A series follows the pattern: 2, 6, 12, 20, 30, ?, 56. What is the missing term?

Accepted Answer

Step 1: Observe differences between consecutive terms: 6 − 2 = 4, 12 − 6 = 6, 20 − 12 = 8, 30 − 20 = 10. Step 2: The first differences are 4, 6, 8, 10 (increasing by 2 each time). Step 3: The next first difference should be 12, so missing term = 30 + 12 = 42. Step 4: Verify: next difference = 14, so 42 + 14 = 56 ✓. Therefore, the missing term is 42.

Question 39

A series is defined as: 1, 4, 9, 16, 25, ?, 49. What is the missing term?

Accepted Answer

Step 1: Observe the terms: 1 = 1², 4 = 2², 9 = 3², 16 = 4², 25 = 5². Step 2: This is a series of perfect squares: aₙ = n². Step 3: The missing term corresponds to n = 6, so missing term = 6² = 36. Step 4: Verify: 7² = 49 ✓. Therefore, the missing term is 36.

Question 40

In a number series, the first term is 5 and each subsequent term is obtained by multiplying the previous term by 2 and then subtracting 3. What is the 5th term of this series?

Accepted Answer

Step 1: Term 1 = 5. Step 2: Term 2 = 5 × 2 − 3 = 10 − 3 = 7. Step 3: Term 3 = 7 × 2 − 3 = 14 − 3 = 11. Step 4: Term 4 = 11 × 2 − 3 = 22 − 3 = 19. Step 5: Term 5 = 19 × 2 − 3 = 38 − 3 = 35. Therefore, the 5th term is 35.

Question 41

In a number series, the first term is 3 and each subsequent term is obtained by multiplying the previous term by 2 and then subtracting 1. What is the 6th term of this series?

Accepted Answer

Step 1: Identify the recurrence relation: a(n) = 2·a(n-1) - 1. Step 2: Calculate term by term. a(1) = 3. a(2) = 2(3) - 1 = 5. a(3) = 2(5) - 1 = 9. a(4) = 2(9) - 1 = 17. a(5) = 2(17) - 1 = 33. a(6) = 2(33) - 1 = 65. Therefore, the 6th term is 65.

Question 42

In a series, the first term is 100 and each term is 80% of the previous term. What is the 4th term (rounded to the nearest integer)?

Accepted Answer

Step 1: This is a geometric series with first term a = 100 and common ratio r = 0.8. Step 2: Term 1 = 100. Step 3: Term 2 = 100 × 0.8 = 80. Step 4: Term 3 = 80 × 0.8 = 64. Step 5: Term 4 = 64 × 0.8 = 51.2 ≈ 51. Therefore, the 4th term is 51.

Question 43

A number series follows the pattern: 3, 6, 12, 24, ?, 96. What is the missing term?

Accepted Answer

Step 1: Observe the ratio between consecutive terms: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. Step 2: This is a geometric series with common ratio r = 2. Step 3: The missing term = 24 × 2 = 48. Step 4: Verify: 48 × 2 = 96 ✓. Therefore, the missing term is 48.

Question 44

A series follows the pattern: 3, 6, 12, 24, 48, ... What is the sum of the 6th and 7th terms?

Accepted Answer

Step 1: Identify the pattern—each term is double the previous term (geometric series with ratio 2). Step 2: 6th term = 48 × 2 = 96. Step 3: 7th term = 96 × 2 = 192. Step 4: Sum = 96 + 192 = 288. Therefore, the answer is 288.

Question 45

In a series, each term is the sum of the two preceding terms: 2, 3, 5, 8, 13, ?, 34. What is the missing term?

Accepted Answer

Step 1: This is a Fibonacci-type series where aₙ = aₙ₋₁ + aₙ₋₂. Step 2: Verify pattern: 2 + 3 = 5 ✓, 3 + 5 = 8 ✓, 5 + 8 = 13 ✓. Step 3: Missing term = 8 + 13 = 21. Step 4: Verify next: 13 + 21 = 34 ✓. Therefore, the missing term is 21.

Question 46

A series is defined as: T(n) = n² + 2n + 1. What is the difference between the 5th and 3rd terms?

Accepted Answer

Step 1: Calculate T(5) = 5² + 2(5) + 1 = 25 + 10 + 1 = 36. Step 2: Calculate T(3) = 3² + 2(3) + 1 = 9 + 6 + 1 = 16. Step 3: Difference = T(5) − T(3) = 36 − 16 = 20. Therefore, the answer is 20.

Agniveer (All) Number Series

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