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Number and Alphabet Series

49 questions

All types49Multiple choice26Numeric13Fill in the blank10
Question 41Question →

Which of the following alphanumeric terms logically completes the given series: B7Z,E14W,J23T,Q34Q,…B7Z, E14W, J23T, Q34Q, \dots?

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Answer: Z47NZ47N

Answer

Z47NZ47N
Each character position follows a distinct rule: the first letters (B,E,J,QB, E, J, Q) advance by +3,+5,+7,+9+3, +5, +7, +9 to reach ZZ; the middle numbers (7,14,23,347, 14, 23, 34) advance by +7,+9,+11,+13+7, +9, +11, +13 to reach 4747; and the last letters (Z,W,T,QZ, W, T, Q) step backward by 33 positions each time to reach NN. Combining these produces the correct completed term Z47NZ47N.

Step-by-Step Solution

1
Analyze the first letter in each term
The positional values of the first letters are B=2,E=5,J=10,Q=17B=2, E=5, J=10, Q=17. The differences between consecutive letters are +3,+5,+7+3, +5, +7. Following this pattern, the next increment is +9+9, so 17+9=2617 + 9 = 26, which corresponds to ZZ.
The difference between consecutive first-letter positions increases by +2+2 at each step.
2
Analyze the middle number in each term
The numbers are 7,14,23,347, 14, 23, 34. The differences are 14−7=714 - 7 = 7, 23−14=923 - 14 = 9, and 34−23=1134 - 23 = 11. The next difference must be +13+13, so 34+13=4734 + 13 = 47.
The numerical pattern follows an increasing sequence of odd numbers starting from 77.
3
Analyze the last letter in each term
The positional values of the last letters are Z=26,W=23,T=20,Q=17Z=26, W=23, T=20, Q=17. The pattern is a constant decrease of 33 positions (26→23→20→1726 \rightarrow 23 \rightarrow 20 \rightarrow 17). The next position is 17−3=1417 - 3 = 14, which corresponds to NN.
A uniform reverse shift of 33 units is applied to the final letter of each term.
4
Combine all three components
Merging the first letter (ZZ), middle number (4747), and last letter (NN) yields the term Z47NZ47N.
All three sub-patterns operate independently to define the next term in the series.

Key Concept

Multi-Pattern Alphanumeric Series Completion
Question 42Question →
Consider the following alphanumeric series:
B4,E9,I19,N39,T79,…B4, E9, I19, N39, T79, \dots
Which of the following terms logically completes the series?
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Answer: A159A159

Answer

The term that logically completes the series is A159A159.
The letter sequence advances by increasing positional shifts (+3, +4, +5, +6, +7), bringing T (20) to 27, which wraps around to A. The number sequence follows nk=2nk−1+1n_{k} = 2n_{k-1} + 1, transforming 79 into 159. Combining these yields A159A159.

Step-by-Step Solution

1
Analyze the pattern governing the letter component of the series.
Positions in the English alphabet: B is 2, E is 5, I is 9, N is 14, T is 20.
Differences between consecutive letter positions:
5−2=+35 - 2 = +3
9−5=+49 - 5 = +4
14−9=+514 - 9 = +5
20−14=+620 - 14 = +6
Following this progression, the next shift must be +7+7.
Adding 7 to the position of T (20) gives 20+7=2720 + 7 = 27. Wrapping around the 26-letter alphabet (27−26=127 - 26 = 1), the 1st letter is A.
The letter pattern increases the positional shift increment by 1 at each step.
2
Analyze the pattern governing the numerical component of the series.
The numerical terms are 4, 9, 19, 39, 79.
Evaluating the relation between consecutive terms:
4×2+1=94 \times 2 + 1 = 9
9×2+1=199 \times 2 + 1 = 19
19×2+1=3919 \times 2 + 1 = 39
39×2+1=7939 \times 2 + 1 = 79
Applying the same rule to 79:
79×2+1=15979 \times 2 + 1 = 159.
Each number is generated by doubling the preceding number and adding 1.
3
Combine the letter and numerical results.
The next term in the series is A159A159.
Joining the identified letter A with the calculated number 159 yields the complete next term.

Key Concept

Alphanumeric Series Completion via Increasing Positional Shift and Recurrence Progression
Question 43Question →

Which of the following alphanumeric terms logically completes the given sequence?

A2P,D9M,I28J,P65G,…A2P, D9M, I28J, P65G, \dots
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Answer: Y126DY126D

Answer

Y126DY126D
The correct term combines three distinct structural rules: the first letter follows perfect square positions (12,22,32,42,52→Y1^2, 2^2, 3^2, 4^2, 5^2 \rightarrow Y), the numerical value follows n3+1n^3 + 1 (53+1=1265^3 + 1 = 126), and the last letter decreases by 3 positions (G−3=DG - 3 = D), giving Y126DY126D.

Step-by-Step Solution

1
Analyze the pattern of the first letter in each term
The alphabetical positions are A=1A=1, D=4D=4, I=9I=9, and P=16P=16. These numbers are perfect squares (12,22,32,421^2, 2^2, 3^2, 4^2). The 5th position is 52=255^2 = 25, which corresponds to the letter YY.
Establishing the sequence rule for the first component.
2
Analyze the pattern of the middle number in each term
The numeric values are 2,9,28,652, 9, 28, 65. Expressing them in terms of term position nn: 13+1=21^3 + 1 = 2, 23+1=92^3 + 1 = 9, 33+1=283^3 + 1 = 28, 43+1=654^3 + 1 = 65. For n=5n=5, the value is 53+1=125+1=1265^3 + 1 = 125 + 1 = 126.
Establishing the algebraic relation governing the middle numeric component.
3
Analyze the pattern of the third letter in each term
The alphabetical positions are P=16P=16, M=13M=13, J=10J=10, and G=7G=7. The pattern decreases by 3 each step (−3-3). The next position is 7−3=47 - 3 = 4, which corresponds to the letter DD.
Establishing the reverse linear shift rule for the third component.
4
Combine the three derived components
Combining the first letter (YY), middle number (126126), and final letter (DD) yields Y126DY126D.
Constructing the final term of the series.

Key Concept

Multi-Pattern Alphanumeric Series
Estimated Time:2m 0s
Question 44Question →

Consider the numerical sequence: 5,11,24,51,106,217,…5, 11, 24, 51, 106, 217, \dots. What is the value of the next term in this sequence?

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Answer: 440

Answer

The next term in the sequence is 440.
The sequence follows the recurrence pattern where each term is twice the previous term plus an increasing integer offset: 5×2+1=115 \times 2 + 1 = 11, 11×2+2=2411 \times 2 + 2 = 24, 24×2+3=5124 \times 2 + 3 = 51, 51×2+4=10651 \times 2 + 4 = 106, 106×2+5=217106 \times 2 + 5 = 217. Following this pattern, the next term is 217×2+6=440217 \times 2 + 6 = 440.

Step-by-Step Solution

1
Examine the relationship between successive terms.
The differences between successive terms are 6,13,27,55,1116, 13, 27, 55, 111, and the second-level differences are 7,14,28,567, 14, 28, 56 (doubling each step). Alternatively, each term is multiplied by 2 and increased by an incrementing integer.
Analyzing both direct operations and higher-order differences establishes the recurrence rule an+1=2an+na_{n+1} = 2a_n + n.
2
Determine the required operation for the next step in the sequence.
Multiply the 6th term (217) by 2 and add 6.
The additive component increases by 1 at each term (+1,+2,+3,+4,+5→+6+1, +2, +3, +4, +5 \rightarrow +6).
3
Calculate the value of the next term.
217×2+6=440217 \times 2 + 6 = 440.
217×2=434217 \times 2 = 434, and adding 6 yields 440.

Key Concept

Recurrent series with multiplicative factor and sequential linear increment
Estimated Time:1m 30s
Question 45Question →
Consider the following alphanumeric sequence:
W7,T14,P26,K45,E73,…W7, T14, P26, K45, E73, \dots
Which of the following options logically completes the sequence?
Show answer & explanation

Answer: X112X112

Answer

The correct option is X112X112, obtained by shifting the letter term backwards by 7 positions (E to X) and adding 39 to the previous numerical term (73 + 39 = 112) based on the double-difference pattern.
The correct answer is X112X112. The letter pattern subtracts increasing integer steps (-3, -4, -5, -6, -7), where E (5) minus 7 wraps to X (24). The numerical pattern relies on a double difference of consecutive odd numbers (+5, +7, +9, +11), adding 39 to 73 to yield 112.

Step-by-Step Solution

1
Analyze the letter progression pattern.
Alphabetical positions: W(23), T(20), P(16), K(11), E(5). The differences are -3, -4, -5, -6. The next shift must be -7.
The step size subtracted from the letter positions increases by 1 at each step.
2
Calculate the next letter term.
Subtracting 7 steps backwards from E (5): 5−7=−2≡24(mod26)5 - 7 = -2 \equiv 24 \pmod{26}, which corresponds to the 24th letter, X.
Cyclic alphabetical wrapping applies when counting backwards past A.
3
Analyze the numerical progression using first and second differences.
First differences: 14−7=714 - 7 = 7, 26−14=1226 - 14 = 12, 45−26=1945 - 26 = 19, 73−45=2873 - 45 = 28. Second differences: 12−7=512 - 7 = 5, 19−12=719 - 12 = 7, 28−19=928 - 19 = 9.
The second differences form a sequence of consecutive odd numbers (+5, +7, +9).
4
Compute the next numerical term.
The next second difference is +11+11. Thus, the next first difference is 28+11=3928 + 11 = 39. Adding this to the last term yields 73+39=11273 + 39 = 112.
Extrapolating the double-difference pattern determines the missing number.

Key Concept

Alphanumeric series completion with variable reverse positional letter shifts and double-difference numerical patterns
Question 46Question →

Complete the given numerical series by identifying the term that logically follows.

Fill in the blanks below

In the numerical sequence 3,7,11,15,19,…3, 7, 11, 15, 19, \dots, the term that completes the blank is .
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Answer

The next term in the sequence is 23.
Each consecutive term in the given series increases by a fixed constant of 44 (3+4=73 + 4 = 7, 7+4=117 + 4 = 11, 11+4=1511 + 4 = 15, 15+4=1915 + 4 = 19). Adding 44 to 1919 gives the correct term, 2323.

Step-by-Step Solution

1
Find the difference between consecutive terms in the sequence.
7−3=47 - 3 = 4, 11−7=411 - 7 = 4, 15−11=415 - 11 = 4, and 19−15=419 - 15 = 4.
Determining the common difference helps uncover the underlying arithmetic rule of the sequence.
2
Add the constant difference of 4 to the last term given (1919).
19+4=2319 + 4 = 23.
Applying the identified rule (+4) yields the logical next term of the series.

Key Concept

Arithmetic Number Series with Constant Difference
Estimated Time:30s
Question 47Question →
Consider the following numerical sequence:
3,11,38,102,227,…3, 11, 38, 102, 227, \dots
What is the value of the next term in this sequence?
Show answer & explanation

Answer: 443

Answer

The next term in the sequence is 443.
Each term increases by the cube of consecutive natural numbers starting from 2: +2³, +3³, +4³, +5³, and +6³. Adding 6³ (which equals 216) to the preceding term 227 yields 443.

Step-by-Step Solution

1
Find the difference between consecutive terms in the given sequence.
The differences are 8, 27, 64, and 125.
Analyzing first-order differences reveals the rate of increase between terms.
2
Identify the logical pattern governing the sequence of differences.
The differences correspond to consecutive perfect cubes: 2³, 3³, 4³, and 5³.
Recognizing that 8 = 2³, 27 = 3³, 64 = 4³, and 125 = 5³ establishes the cubic growth rule.
3
Calculate the next term by extending the identified pattern.
Next difference = 6³ = 216, resulting in next term = 227 + 216 = 443.
Following the consecutive cubic sequence, the fifth difference must be 6³.

Key Concept

Cube-Difference Series Progression
Question 48Question →

Determine the missing values to logically complete the given numerical sequence.

Fill in the blanks below

In the sequence $3, 8, 18, 38, , 158, $, the first missing term is and the second missing term is .
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Answer

The first missing term is 78 and the second missing term is 318.
The pattern governing the sequence can be defined recursively as an+1=2an+2a_{n+1} = 2a_n + 2. Following 3838, the term is (38×2)+2=78(38 \times 2) + 2 = 78. Following 158158, the term is (158×2)+2=318(158 \times 2) + 2 = 318. Alternatively, observing the difference between consecutive terms shows +5,+10,+20,+40,+80,+160+5, +10, +20, +40, +80, +160, yielding 38+40=7838 + 40 = 78 and 158+160=318158 + 160 = 318.

Step-by-Step Solution

1
Analyze the rule connecting consecutive terms in the sequence
Each term is generated by multiplying the preceding term by 22 and adding 22: (3×2)+2=8(3 \times 2) + 2 = 8, (8×2)+2=18(8 \times 2) + 2 = 18, and (18×2)+2=38(18 \times 2) + 2 = 38. Alternatively, the differences between successive terms double each step (+5,+10,+20,…+5, +10, +20, \dots).
Identifying the mathematical operational relationship allows computing unknown terms.
2
Calculate the value for the first blank following 3838
(38×2)+2=76+2=78(38 \times 2) + 2 = 76 + 2 = 78 (or 38+40=7838 + 40 = 78).
Applying the identified rule to 3838 yields the first missing term.
3
Verify the calculated value against the given term 158158
(78×2)+2=156+2=158(78 \times 2) + 2 = 156 + 2 = 158 (or 78+80=15878 + 80 = 158).
Ensures consistency across the pattern before proceeding to the final term.
4
Calculate the value for the second blank following 158158
(158×2)+2=316+2=318(158 \times 2) + 2 = 316 + 2 = 318 (or 158+160=318158 + 160 = 318).
Applying the operational rule to 158158 yields the second missing term.

Key Concept

Mixed Arithmetic Series and Double Difference Progression
Estimated Time:1m 15s
Question 49Question →

A cryptography algorithm generates a continuous stream of security tokens based on a polynomial progression. The first five tokens generated by the system are 55, 1212, 3131, 6868, and 129129.

What is the numerical value of the sixth token generated by this algorithm?

Show answer & explanation

Answer: 220

Answer

220
The sequence follows the mathematical rule where the nn-th term is equal to n3+4n^3 + 4. The first term is 13+4=51^3 + 4 = 5, the second is 23+4=122^3 + 4 = 12, up to the fifth which is 53+4=1295^3 + 4 = 129. Applying this rule to the sixth position yields 63+4=216+4=2206^3 + 4 = 216 + 4 = 220. Alternatively, solving via successive differences confirms that the constant third difference is 6, which accurately points to 220.

Step-by-Step Solution

1
Calculate the first-level differences between the given consecutive tokens.
The sequence of differences is 7, 19, 37, and 61.
Establishing the initial rate of change helps identify if a linear or higher-order polynomial pattern exists.
2
Calculate the second-level differences from the results of Step 1.
The differences between the differences are 12, 18, and 24.
Finding the differences of the differences reveals simpler underlying arithmetic progressions.
3
Identify the pattern in the second-level differences and extrapolate the next value.
The values (12, 18, 24) increase by exactly 6 each time. The next second-level difference is 24 + 6 = 30.
Extending this constant third-level difference (+6) is required to build the sequence forward.
4
Calculate the next first-level difference and the final sequence term.
Next first-level difference = 61 + 30 = 91. Next sequence term = 129 + 91 = 220.
Applying the extrapolated values back up the chain yields the target term in the main sequence.

Key Concept

Number series completion using the method of successive differences or cube offsets.
Estimated Time:1m 15s
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