Number Series Questions and Answers

Answer: C

All three statements are correct.

Each row is a geometric progression with common ratio 3, and the second column gives the next term.

• 2, 6, 18: next is \(18 \times 3 = 54\).
• 3, 9, 27: next is \(27 \times 3 = 81\).
• 5, 15, 45: next is \(45 \times 3 = 135\).

Every row continues its own series correctly, so no row is wrong. A solver who reads row 2 as powers of 3 still gets 81, so the answer does not depend on which rule is spotted.

Answer: B

The next number is 102.

The differences between terms shrink by 6 each time:

\[ \begin{aligned} 141 - 122 &= 19 \\ 122 - 109 &= 13 \\ 109 - 7 &= 102 \end{aligned} \]

The differences 19, 13, 7 form a pattern falling by 6. 105, 104 and 106 would need differences of 4, 5 and 3, which break it.

Answer: C

The 7th term is 288.

The terms come from the prime numbers 2, 3, 5, 7, ... squared, alternately minus and plus 1:

\[ \begin{aligned} 2^{2} - 1 &= 3,\quad 3^{2} + 1 = 10 \\ 5^{2} - 1 &= 24,\quad 7^{2} + 1 = 50 \end{aligned} \]

Continuing, \(11^{2} - 1 = 120\), \(13^{2} + 1 = 170\), and the 7th term uses the 7th prime, 17: \(17^{2} - 1 = 288\). 290 is \(17^{2} + 1\), with the sign pattern broken, and 224 and 226 do not fit.

Answer: B

The next number is 1872.

After the opening 1, 1 the series repeats a cycle of three operations: add the last two terms, square the result, then multiply the last two terms.

\[ \begin{aligned} 1 + 1 &= 2, \quad 2^{2} = 4, \quad 2 \times 4 = 8 \\ 4 + 8 &= 12, \quad 12^{2} = 144 \\ 12 \times 144 &= 1728 \end{aligned} \]

The next step in the cycle is addition again: \(144 + 1728 = 1872\).

The other options do not come from any of the three operations applied to 144 and 1728: their product and the square of 1728 are far larger, and no step of the cycle subtracts.

Answer: A

The next term is 2744.

Each term is the cube of an even number, rising by 2 each time:

\[ \begin{aligned} 216 &= 6^3,\ 512 = 8^3 \\ 1000 &= 10^3,\ 1728 = 12^3 \\ \text{next} &= 14^3 = 2744 \end{aligned} \]

The differences (296, 488, 728) grow too, but not in a way that points to 2352 or 2016; those come from guessing a simple pattern in the differences. Since 2744 is listed, 'None of these' is wrong.

Answer: B

The missing number is 64.

The series is the squares of the numbers from 12 downwards: \(12^{2} = 144\), \(11^{2} = 121\), \(10^{2} = 100\), \(9^{2} = 81\), then \(8^{2} = 64\), followed by \(7^{2} = 49\) and \(6^{2} = 36\).

The differences confirm it: 23, 21, 19, 17, 15, 13, falling by 2 each time. 81 minus 17 is 64, and 64 minus 15 is 49.

72, 76 and 66 are not perfect squares and break the pattern of differences.

Answer: B

There are 4 such 5s.

Check each 5 in 1 3 5 5 4 5 5 7 3 3 5 5 5 2 8 7 2 5 3 and look at its two neighbours:

• 3rd digit (3, 5, 5): both odd, counted.
• 4th digit (5, 5, 4): right is even.
• 6th digit (4, 5, 5): left is even.
• 7th digit (5, 5, 7): both odd, counted.
• 11th digit (3, 5, 5): both odd, counted.
• 12th digit (5, 5, 5): both odd, counted.
• 13th digit (5, 5, 2): right is even.
• 18th digit (2, 5, 3): left is even.

That gives 4. The answer 8 is simply the total number of 5s in the series, the trap for anyone who ignores the condition.

Answer: B

The next term is 34.

The differences between successive terms are consecutive odd numbers:

\[ \begin{aligned} 2 - (-1) &= 3,\ 7 - 2 = 5 \\ 14 - 7 &= 7,\ 23 - 14 = 9 \end{aligned} \]

The next difference is 11, so the missing term is 23 + 11 = 34. Equivalently, the terms are \(n^2 - 2\) for n = 1, 2, 3 and so on, and \(6^2 - 2 = 34\). 36 and 35 come from adding 13 or 12 instead of 11.

The formula checks every given term: 1 - 2 = -1, 4 - 2 = 2, 9 - 2 = 7, 16 - 2 = 14 and 25 - 2 = 23. Option C, 32, repeats the difference of 9.

Answer: D

The missing number is 65.

Each term is one more than a cube:

\[ \begin{aligned} 2 &= 1^{3} + 1,\quad 9 = 2^{3} + 1,\quad 28 = 3^{3} + 1 \\ 65 &= 4^{3} + 1,\quad 126 = 5^{3} + 1 \end{aligned} \]

so the gap is filled by \(4^{3} + 1 = 65\). 63 is \(4^{3} - 1\), and 57 and 53 do not fit the pattern; the differences 7, 19, 37, 61 also confirm 65, since they rise by 12, 18, 24.

Answer: C

The missing number is 20.

Each term is the product of two consecutive numbers:

\[ \begin{aligned} &1 \times 2,\ 2 \times 3,\ 3 \times 4, \\ &4 \times 5,\ 5 \times 6 \end{aligned} \]

which gives 2, 6, 12, 20, 30. Equivalently, the differences grow by 2 each time: 4, 6, 8, 10, so the gap after 12 must be 8 and the gap before 30 must be 10. Each wrong option breaks this: 15 would leave gaps of 3 and 15, 18 gaps of 6 and 12, and 25 gaps of 13 and 5.