Number Series Questions and Answers

Answer: C

The answer is \(43^{2}\).

In each line the three numbers are \(n\), \(n + 1\) and their product \(n(n + 1)\), and the right-hand side is the square of (product + 1):

\[ \begin{aligned} 1, 2, 2 &\to 2 + 1 = 3 \\ 2, 3, 6 &\to 6 + 1 = 7 \\ 3, 4, 12 &\to 12 + 1 = 13 \\ 6, 7, 42 &\to 42 + 1 = 43 \end{aligned} \]

Check: 36 + 49 + 1764 = 1849 = \(43^{2}\). \(42^{2}\) forgets the +1, and \(45^{2}\) and \(49^{2}\) do not fit the identity.

Answer: B

The unknown number is 12.

In each row the first number minus the second gives the third:

\[ \begin{aligned} 20 - 13 &= 7 \\ 30 - 8 &= 22 \\ 40 - ? &= 28 \Rightarrow ? = 12 \end{aligned} \]

Equivalently the second and third numbers add up to the first (13 + 7 = 20, 8 + 22 = 30). 10, 16 and 20 do not satisfy the relation in the third row.

Answer: D

The successive differences of 0.5, 0.6, 1.5, 4, 8.9 are 0.1, 0.9, 2.5, 4.9, which are \(0.1\times1^{2}\), \(0.1\times3^{2}\), \(0.1\times5^{2}\) and \(0.1\times7^{2}\), the squares of the odd numbers.

The next difference is \(0.1\times9^{2}=8.1\), so the next term is 8.9 + 8.1 = 17.0, option D. The values 12.6, 13.5 and 14.3 do not follow the pattern.

A series whose differences are squares of odd numbers scaled by 0.1 is a typical aptitude pattern, and the check 0.5 + 0.1 = 0.6, 0.6 + 0.9 = 1.5, 1.5 + 2.5 = 4.0, 4.0 + 4.9 = 8.9 confirms it.

Answer: C

The wrong number is 48.

Each term is obtained from the one before it by subtracting 4 and then halving:

\[ \begin{aligned} (380 - 4) \div 2 &= 188 \\ (188 - 4) \div 2 &= 92 \\ (92 - 4) \div 2 &= 44 \\ (44 - 4) \div 2 &= 20 \\ (20 - 4) \div 2 &= 8 \\ (8 - 4) \div 2 &= 2 \end{aligned} \]

The term after 92 should be 44, not 48. Every other term, including 20, 8 and 2, follows from 44 by the same rule, so 20, 92 and 2 are all correct members of the series.

Answer: B

The missing term is 120.

Each term is the previous term multiplied by the next counting number, so the terms are factorials:

\[ \begin{aligned} &1,\ 1 \times 1 = 1,\ 1 \times 2 = 2,\ 2 \times 3 = 6 \\ &6 \times 4 = 24,\ 24 \times 5 = 120,\ 120 \times 6 = 720 \end{aligned} \]

That is \(0!, 1!, 2!, 3!, 4!, 5!, 6!\), and \(5! = 120\). The last term confirms it, since \(120 \times 6 = 720\). 100 and 140 do not lead to 720 by any consistent rule (720 is not a whole multiple of 140), and 13 breaks the steady growth of the series.

Answer: A

The missing numbers are 13 and 49, giving 7, 13, 25, 49, 97.

Each term is twice the previous term minus one:

\[ \begin{aligned} 7 \times 2 - 1 &= 13 \\ 13 \times 2 - 1 &= 25 \\ 25 \times 2 - 1 &= 49 \\ 49 \times 2 - 1 &= 97 \end{aligned} \]

Equivalently, the gaps between terms double each time: 6, 12, 24, 48. The pairs in the other options break the pattern at once; for example 17 would have to follow 7 and lead to 25, but 17 times 2 minus 1 is 33, not 25.

Answer: C

The next number is 309.

Look at the differences between terms:

\[ \begin{aligned} 127 - 114 &= 13,\ 153 - 127 = 26 \\ 192 - 153 &= 39,\ 244 - 192 = 52 \end{aligned} \]

The differences are 13, 26, 39, 52, the multiples of 13, so the next difference is 65 and the next term is \(244 + 65 = 309\). 284 adds only 40, while 344 and 352 add 100 and 108, breaking the pattern of multiples of 13.

Answer: B

7 maps to 56.

Each number is multiplied by the next whole number:

\[ \begin{aligned} 4 \times 5 &= 20,\ 5 \times 6 = 30 \\ 6 \times 7 &= 42,\ 7 \times 8 = 56 \end{aligned} \]

Equivalently, \(n \rightarrow n^2 + n\). 49 is \(7^2\) alone, without the added 7; 63 is \(7 \times 9\) and 70 is \(7 \times 10\), which use the wrong second factor.

Answer: C

The missing number is 867.

The differences between terms increase by 1 each time:

\[ \begin{aligned} 832 - 816 &= 16 \\ 849 - 832 &= 17 \\ 849 + 18 &= 867 \\ 867 + 19 &= 886 \end{aligned} \]

The check with the last term, 886, confirms it: the gaps run 16, 17, 18, 19. 866 and 869 are one or two away, and would leave a gap to 886 of 20 or 17, which breaks the rising pattern, so 'None of these' is wrong too.

Answer: B

The next number is 42.

Each term is the product of two consecutive numbers:

\[ \begin{aligned} 1 \times 2 &= 2,\ 2 \times 3 = 6 \\ 3 \times 4 &= 12,\ 4 \times 5 = 20 \\ 5 \times 6 &= 30,\ 6 \times 7 = 42 \end{aligned} \]

Equivalently, the differences are 4, 6, 8, 10, so the next difference is 12 and \(30 + 12 = 42\). The other options do not follow either pattern.