Number Series Questions and Answers

Answer: D

The terms are the cubes of 3, 4, 5 and 6: \(27 = 3^3\), \(64 = 4^3\), \(125 = 5^3\) and \(216 = 6^3\). The next term is the cube of 7.

\[ 7^3 = 343 \]

The values 340, 341 and 342 are close to 343 but are not cubes. The answer is 343, option D.

As a check, the gaps between terms are 37, 61 and 91, which increase by 24 and 30, so the next gap is 36 more than 91, that is 127, and 216 + 127 = 343, which agrees with the cube of 7.

Answer: B

The next term is 218. Each term is double the previous term plus a number that rises by one each time: 12 x 2 + 1 = 25, 25 x 2 + 2 = 52, 52 x 2 + 3 = 107. The next step is 107 x 2 + 4 = 218. The other options, 216, 219 and 220, would need an added number of 2, 5 or 6, which breaks the sequence of +1, +2, +3, +4. The differences between the terms are 13, 27, 55, and each one equals the previous difference doubled plus 1, since 13 x 2 + 1 = 27 and 27 x 2 + 1 = 55. Both the terms and the differences follow the same rule, so the next difference is 111 and the term is 107 + 111 = 218.

Answer: B

The next number is 300. The differences between the terms are 500 - 430 = 70, 430 - 380 = 50, 380 - 345 = 35 and 345 - 320 = 25. These differences themselves fall by 20, 15 and 10, so the next fall is 5 and the next difference is 25 - 5 = 20. Then 320 - 20 = 300. The other options, 290, 305 and 310, would need a difference of 30, 15 or 10, which does not continue the pattern.

Answer: A

The next number is 99999. The terms are 10 - 1 = 9, 100 + 1 = 101, 1000 - 1 = 999, 10000 + 1 = 10001, so the powers of ten go up by one each time and the added number alternates between -1 and +1. The fifth term is \(10^{5} - 1 = 99999\).

99998 and 99990 break the pattern of a power of ten plus or minus one, and 100000 has the right power but the wrong sign.

Check by the pattern of the differences from powers of ten: 9 is 10 minus 1, 101 is 100 plus 1, 999 is 1000 minus 1 and 10001 is 10000 plus 1. The sign alternates minus, plus, minus, plus, so the next sign is minus and the next power of ten is 100000, which gives 99999.

Answer: B

The missing number is 960. Take the terms in odd places and in even places separately. The odd places hold 128, 256, 1024. The even places hold 120, 240 and the missing term. Each even-place term is \(\frac{15}{16}\) of the term just before it: 120 is \(\frac{15}{16}\) of 128, and 240 is \(\frac{15}{16}\) of 256.

480 and 512 are not 15/16 of 1024 and 1000 does not fit that ratio either, so only 960 continues the pattern.

Answer: B

The differences between the terms are 120 - 90 = 30, 156 - 120 = 36, 198 - 156 = 42 and 246 - 198 = 48. These differences themselves increase by 6 each time, so the next difference is 54, and the next term is 246 + 54 = 300. The answer is 300, option (B). The option 350 would need a difference of 104, and 296 and 306 do not follow the rise of 6 in the differences, which are 6, 6 and 6 for the earlier terms. Series in which the difference between the terms goes up by a fixed amount are called second-order series, and the same working is used in many number series questions.

Answer: B

The missing term is 230. The differences between successive terms are 12, 24, 36 and 48, so each difference is 12 more than the one before. The next difference is therefore 60.

\[ 62-50=12,\ 86-62=24,\ 122-86=36,\ 170-122=48 \]

\[ 170+60=230 \]

The option 225 would need a difference of 55, 240 a difference of 70 and 250 a difference of 80. None of these continues the steady rise of 12, so only 230 fits the series.

Answer: A

The next term is 169.

The terms are the squares of consecutive prime numbers: 2² = 4, 3² = 9, 5² = 25, 7² = 49 and 11² = 121. The next prime is 13, and 13² = 169.

196 is 14 squared, and 14 is not a prime. 179 and 324 are not squares of primes either.

The terms 4, 9, 25, 49 and 121 are the squares of 2, 3, 5, 7 and 11, which are consecutive primes. The prime after 11 is 13, because 12 is even. Squaring it gives 169. The number 196 is the square of 14, which is not prime, so it breaks the pattern.

Answer: A

The missing number is 113. The series adds a square and a cube alternately: \[ 6 + 1 = 7 \] \[ 7 + 2^{3} = 15 \] \[ 15 + 3^{2} = 24 \] \[ 24 + 4^{3} = 88 \] \[ 88 + 5^{2} = 113 \] \[ 113 + 6^{3} = 329 \] The last step reaches the given 329 exactly, which confirms the pattern. The other options 114, 115 and 116 would leave a gap that is not 216, so the final cube would not fit. Checking the last step shows that 113 + 216 = 329, which agrees with the number given in the series.

Answer: D

The missing number is 1785. Each term is three times the previous term minus 6: 25 x 3 - 6 = 69, 69 x 3 - 6 = 201, 201 x 3 - 6 = 597, so the next is 597 x 3 - 6 = 1785.

1775, 1765 and 1825 do not follow from 597 under the rule that fits all four earlier steps.

Check the pattern on the earlier terms: 25 x 3 - 6 = 69, 69 x 3 - 6 = 201 and 201 x 3 - 6 = 597, so one rule explains every step and gives 1785.