Number Series Questions in English

(D) The given series is an alternating series consisting of two interleaved sequences.
Sequence $1$: $15, 16, 17, ...$
Here,each term increases by $1$ ($15 + 1 = 16$,$16 + 1 = 17$,$17 + 1 = 18$).
Sequence $2$: $18, 19, 20, ...$
Here,each term increases by $1$ ($18 + 1 = 19$,$19 + 1 = 20$).
The series follows the pattern: $15 (+3), 18 (-2), 16 (+3), 19 (-2), 17 (+3), 20 (-2), ?$
Following the pattern,the next term after $20$ is $17 + 3 = 20$ (if continuing sequence $1$) or $20 - 2 = 18$ (if continuing sequence $2$).
However,looking at the alternating pattern: $15, 18, 16, 19, 17, 20, 18$.
The next term is $18$.

(A) To find the next number in the series,we observe the pattern between consecutive terms:
$1050 \div 2.5 = 420$
$420 \div 2.5 = 168$
$168 \div 2.5 = 67.2$
$67.2 \div 2.5 = 26.88$
$26.88 \div 2.5 = 10.752$
Following this pattern,the next term is:
$10.752 \div 2.5 = 4.3008$
Therefore,the correct option is $A$.

(D) The given series is $0, 6, 24, 60, 120, 210, ?$.
This series can be expressed as $n^3 - n$ for $n = 1, 2, 3, 4, 5, 6, 7, ...$:
For $n=1: 1^3 - 1 = 0$
For $n=2: 2^3 - 2 = 8 - 2 = 6$
For $n=3: 3^3 - 3 = 27 - 3 = 24$
For $n=4: 4^3 - 4 = 64 - 4 = 60$
For $n=5: 5^3 - 5 = 125 - 5 = 120$
For $n=6: 6^3 - 6 = 216 - 6 = 210$
For $n=7: 7^3 - 7 = 343 - 7 = 336$
Therefore,the next number is $336$.

(C) The pattern in the series is based on the addition of consecutive powers of $2$ or doubling the difference between consecutive terms.
Step $1$: $9 - 7 = 2$
Step $2$: $13 - 9 = 4$
Step $3$: $21 - 13 = 8$
Step $4$: $37 - 21 = 16$
The differences are $2, 4, 8, 16$,which follow the pattern $2^1, 2^2, 2^3, 2^4$.
The next difference should be $2^5 = 32$.
Therefore,the next term is $37 + 32 = 69$.

(C) The pattern in the series is determined by subtracting consecutive powers of $2$ in the denominator of the difference,or simply halving the difference each time:
$36 - 8 = 28$
$28 - 4 = 24$
$24 - 2 = 22$
$22 - 1 = 21$
Thus,the next number in the series is $21$.

(D) The given series is $0, 4, 18, 48, ?, 180$.
We can observe the pattern by finding the differences between consecutive terms:
$4 - 0 = 4$
$18 - 4 = 14$
$48 - 18 = 30$
The differences are $4, 14, 30, ...$
Now,find the second-level differences:
$14 - 4 = 10$
$30 - 14 = 16$
The difference between these is $16 - 10 = 6$.
Following this pattern,the next second-level difference should be $16 + 6 = 22$.
So,the next first-level difference should be $30 + 22 = 52$.
Therefore,the missing term is $48 + 52 = 100$.
Alternatively,the series follows the pattern $n^2(n-1)$ for $n = 1, 2, 3, 4, 5, 6$:
For $n=1: 1^2(0) = 0$
For $n=2: 2^2(1) = 4$
For $n=3: 3^2(2) = 18$
For $n=4: 4^2(3) = 48$
For $n=5: 5^2(4) = 100$
For $n=6: 6^2(5) = 180$
Thus,the missing term is $100$.

(B) The relationship is based on the alphabetical position of the digits where $9$ corresponds to $I$ (the $9^{th}$ letter),$8$ corresponds to $H$ (the $8^{th}$ letter),and $7$ corresponds to $G$ (the $7^{th}$ letter).
Following the same pattern for $654$:
$6$ corresponds to $F$ (the $6^{th}$ letter).
$5$ corresponds to $E$ (the $5^{th}$ letter).
$4$ corresponds to $D$ (the $4^{th}$ letter).
Therefore,$654$ corresponds to $FED$.

(D) The pattern follows the relationship $(n^{2} - 1) : (n^{3} + 1)$.
For the first pair: $5^{2} - 1 = 25 - 1 = 24$ and $5^{3} + 1 = 125 + 1 = 126$.
For the second pair,we observe $48 = 7^{2} - 1$.
Following the same logic,the next term is $7^{3} + 1 = 343 + 1 = 344$.
Therefore,the correct option is $D$.

(D) The given pattern is based on the cubes of consecutive natural numbers.
$(1)^3 = 1$
$(2)^3 = 8$
Following the same pattern for the second pair:
$(3)^3 = 27$
$(4)^3 = 64$
Therefore,the missing number is $64$.

(A) Analyze the differences between consecutive terms in the sequence:
$142 - 119 = 23$
$119 - 100 = 19$
$100 - 83 = 17$
$83 - 65 = 18$ (This does not follow the pattern of prime numbers: $23, 19, 17, 13, 11, 7$)
If we subtract $13$ from $83$,we get $83 - 13 = 70$.
Then,$70 - 11 = 59$ and $59 - 7 = 52$.
Since $70$ fits the pattern and $65$ does not,the odd number in the sequence is $65$.

(C) Analyze the pattern of the sequence: $8, 12, 24, 46, 72, 108, 152$.
Step $1$: $8 \times 1.5 = 12$
Step $2$: $12 \times 2 = 24$
Step $3$: $24 \times 1.5 = 36$ (The given number is $46$,which is incorrect).
Step $4$: $36 \times 2 = 72$
Step $5$: $72 \times 1.5 = 108$
Step $6$: $108 \times 2 = 216$ (The given number is $152$,which is also incorrect based on the pattern).
However,looking at the sequence $8, 12, 24, 46, 72, 108, 152$,the number $46$ is the first point of deviation from the pattern $\times 1.5, \times 2, \times 1.5, \times 2, \dots$
Therefore,the odd number is $46$.

(C) Let us analyze the differences between consecutive terms in the sequence:
$25 - 13 = 12$
$40 - 25 = 15$
$57 - 40 = 17$
$79 - 57 = 22$
$103 - 79 = 24$
$130 - 103 = 27$
Observing the pattern of differences: $12, 15, 18, 21, 24, 27$ (multiples of $3$).
If we follow this pattern:
$13 + 12 = 25$
$25 + 15 = 40$
$40 + 18 = 58$ (instead of $57$)
$58 + 21 = 79$
$79 + 24 = 103$
$103 + 27 = 130$
Thus,the odd number in the sequence is $57$.

(D) Analyze the pattern of the sequence: $2, 10, 18, 54, 162, 486, 1458$.
Starting from the third term,each term is obtained by multiplying the previous term by $3$.
$18 \times 3 = 54$
$54 \times 3 = 162$
$162 \times 3 = 486$
$486 \times 3 = 1458$
If we apply this rule backwards from $18$,the term before $18$ should be $18 / 3 = 6$.
However,the sequence provides $10$ instead of $6$.
Therefore,$10$ is the odd number in the sequence.

(A) Let us analyze the pattern of the sequence:
$850 - 200 = 650$
$650 - 100 = 550$
$550 - 50 = 500$
$500 - 25 = 475$
$475 - 12.5 = 462.5$
$462.5 - 6.25 = 456.25$
In the given sequence,the second term is $600$,but according to the pattern of subtracting half of the previous difference,it should be $650$. Therefore,$600$ is the odd number in the sequence.

(A) The pattern of the series is as follows:
$12 \times 1 = 12$
$12 \times 1.5 = 18$
$18 \times 2 = 36$
$36 \times 2.5 = 90$
$90 \times 3 = 270$
Following this pattern,the next multiplier is $3.5$.
$270 \times 3.5 = 945$
Therefore,the next number in the series is $945$.

(D) The pattern followed in the series is:
$1015 \div 2 + 0.5 = 508$
$508 \div 2 + 1 = 255$
$255 \div 2 + 1.5 = 129$
$129 \div 2 + 2 = 66.5$
$66.5 \div 2 + 2.5 = 35.75$
$35.75 \div 2 + 3 = 20.875$
Thus,the missing number is $35.75$.

(C) The pattern in the series is as follows:
$8 \times 1 + 1 = 9$
$9 \times 2 + 2 = 20$
$20 \times 3 + 3 = 63$
$63 \times 4 + 4 = 256$
$256 \times 5 + 5 = 1285$
Following this pattern,the next term is:
$1285 \times 6 + 6 = 7710 + 6 = 7716$

(D) The pattern followed in the series is: divide the previous number by $2$ and then subtract $6$ from the result.
$980 \div 2 - 6 = 490 - 6 = 484$
$484 \div 2 - 6 = 242 - 6 = 236$
$236 \div 2 - 6 = 118 - 6 = 112$
$112 \div 2 - 6 = 56 - 6 = 50$
$50 \div 2 - 6 = 25 - 6 = 19$
$19 \div 2 - 6 = 9.5 - 6 = 3.5$
Thus,the missing number is $19$.

(B) The pattern followed in the sequence is: divide by $2$ and subtract $2$ $(x_{n+1} = \frac{x_n}{2} - 2)$.
$484 \div 2 - 2 = 242 - 2 = 240$
$240 \div 2 - 2 = 120 - 2 = 118$
$118 \div 2 - 2 = 59 - 2 = 57$
$57 \div 2 - 2 = 28.5 - 2 = 26.5$
$26.5 \div 2 - 2 = 13.25 - 2 = 11.25$
$11.25 \div 2 - 2 = 5.625 - 2 = 3.625$
Comparing this with the given sequence,the term $120$ is incorrect as it should be $118$.

(D) The pattern of the sequence is as follows:
$3 \times 1 + 2 = 5$
$5 \times 2 + 3 = 13$
$13 \times 3 + 4 = 43$
$43 \times 4 + 5 = 177$ (but the given number is $176$)
$177 \times 5 + 6 = 891$
$891 \times 6 + 7 = 5353$
Since the term $176$ does not follow the pattern and should be $177$,the odd number in the sequence is $176$.

(D) The sequence follows the pattern of adding consecutive odd squares:
$6 + 1^2 = 6 + 1 = 7$
$7 + 3^2 = 7 + 9 = 16$
$16 + 5^2 = 16 + 25 = 41$
$41 + 7^2 = 41 + 49 = 90$
$90 + 9^2 = 90 + 81 = 171$
$171 + 11^2 = 171 + 121 = 292$
Comparing this with the given sequence,the term $154$ is incorrect as it should be $171$.

(A) The pattern of the sequence is as follows:
$5 \times 1 + 1^2 = 6$
$6 \times 2 + 2^2 = 16$
$16 \times 3 + 3^2 = 57$
$57 \times 4 + 4^2 = 244$
$244 \times 5 + 5^2 = 1245$
$1245 \times 6 + 6^2 = 7506$
In the given sequence,the second term is $7$,but according to the pattern,it should be $6$. Therefore,$7$ is the odd (wrong) number.

(C) The pattern of the sequence is as follows:
$4 \times 0.5 + 0.5 = 2.5$
$2.5 \times 1 + 1 = 3.5$
$3.5 \times 1.5 + 1.5 = 6.75$
$6.75 \times 2 + 2 = 15.5$
$15.5 \times 2.5 + 2.5 = 41.25$
$41.25 \times 3 + 3 = 126.75$
Comparing this with the given sequence $4, 2.5, 3.5, 6.5, 15.5, 41.25, 126.75$,we can see that $6.5$ is the wrong number,and it should be $6.75$.

(A) Let us analyze the differences between consecutive terms in the sequence:
$34 - 32 = 2$
$37 - 34 = 3$
$46 - 37 = 9$
$62 - 46 = 16$
$87 - 62 = 25$
$123 - 87 = 36$
Observing the differences: $2, 3, 9, 16, 25, 36$.
We can see that $9, 16, 25, 36$ are squares of $3^2, 4^2, 5^2, 6^2$ respectively.
To maintain this pattern,the first two differences should be $1^2 = 1$ and $2^2 = 4$.
If we replace $34$ with $33$,the sequence becomes $32, 33, 37, 46, 62, 87, 123$.
The differences would then be:
$33 - 32 = 1 = 1^2$
$37 - 33 = 4 = 2^2$
$46 - 37 = 9 = 3^2$
$62 - 46 = 16 = 4^2$
$87 - 62 = 25 = 5^2$
$123 - 87 = 36 = 6^2$
Thus,the odd number in the sequence is $34$.

(C) Let us analyze the differences between consecutive terms in the sequence:
$18 - 7 = 11 = 1 \times 11$
$40 - 18 = 22 = 2 \times 11$
$106 - 40 = 66 = 6 \times 11$
$183 - 106 = 77 = 7 \times 11$
$282 - 183 = 99 = 9 \times 11$
$403 - 282 = 121 = 11 \times 11$
The pattern of the differences is $1 \times 11, 2 \times 11, 3 \times 11, 4 \times 11, 5 \times 11, 6 \times 11, 7 \times 11, 8 \times 11, 9 \times 11, 10 \times 11, 11 \times 11$.
If we follow the pattern $18 + (2 \times 11) = 40$ is incorrect because the next term should be $18 + (2 \times 11) = 40$ and $40 + (3 \times 11) = 73$.
Actually,the sequence follows the addition of consecutive multiples of $11$: $11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121$.
$7 + 11 = 18$
$18 + 22 = 40$
$40 + 33 = 73$ (not $106$)
$73 + 44 = 117$ (not $183$)
Wait,let's re-examine: $7, 18, 40, 106, 183, 282, 403$.
The differences are $11, 22, 66, 77, 99, 121$.
This implies $40$ is the odd one out because the sequence should be $7, 18, 40, 73, 117, 172, 238, 315, 403$ is not matching.
Correcting the sequence: $7, 18, 40, 73, 117, 172, 238$. The given sequence has $40$ as the incorrect term relative to the progression of differences.

(D) Let us analyze the differences between consecutive terms in the sequence:
$850 - 843 = 7$
$843 - 829 = 14$
$829 - 808 = 21$
$808 - 788 = 20$
$788 - 745 = 43$
$745 - 703 = 42$
The pattern of differences should be multiples of $7$ $(7, 14, 21, 28, 35, 42)$.
Following this pattern:
$850 - 7 = 843$
$843 - 14 = 829$
$829 - 21 = 808$
$808 - 28 = 780$
$780 - 35 = 745$
$745 - 42 = 703$
Comparing this with the given sequence,the term $788$ is incorrect and should be $780$.

(D) Analyze the differences between consecutive terms in the sequence:
$321 - 33 = 288$
$465 - 321 = 144$
$537 - 465 = 72$
$573 - 537 = 36$
$590 - 573 = 17$
$600 - 590 = 10$
The pattern of differences is $288, 144, 72, 36, 18, 9$ (each term is half of the previous one).
Applying this pattern starting from $573$:
$573 + 18 = 591$
$591 + 9 = 600$
Since $590$ does not fit the pattern,it is the odd number in the sequence.

(A) Let us analyze the pattern of the sequence: $37, 47, 52, 67, 87, 112, 142$.
Calculate the differences between consecutive terms:
$47 - 37 = 10$
$52 - 47 = 5$
$67 - 52 = 15$
$87 - 67 = 20$
$112 - 87 = 25$
$142 - 112 = 30$
Observing the differences: $10, 5, 15, 20, 25, 30$. The pattern should be multiples of $5$ starting from $5$ $(5, 10, 15, 20, 25, 30)$.
If we replace $47$ with $42$,the sequence becomes $37, 42, 52, 67, 87, 112, 142$.
The differences would be: $42-37=5$,$52-42=10$,$67-52=15$,$87-67=20$,$112-87=25$,$142-112=30$.
Thus,$47$ is the odd (incorrect) number in the sequence.

(C) The given series is $586, 587, 586, 581, 570, ?, 522$.
Let's analyze the differences between consecutive terms:
$587 - 586 = +1$
$586 - 587 = -1$
$581 - 586 = -5$
$570 - 581 = -11$
Now,let's look at the differences of these differences:
$-1 - (+1) = -2$
$-5 - (-1) = -4$
$-11 - (-5) = -6$
The pattern of the second difference is $-2, -4, -6, \dots$,which means the next difference should be $-8$.
So,the next difference in the first series should be $-11 + (-8) = -19$.
Therefore,the missing term is $570 - 19 = 551$.
Checking the next term: $551 - 522 = 29$,which is $-19 + (-10) = -29$. The pattern holds.

(D) The given series follows an alternating pattern of subtracting $10, 20, 30$ and adding $15, 25, 35$.
$64 - 10 = 54$
$54 + 15 = 69$
$69 - 20 = 49$
$49 + 25 = 74$
$74 - 30 = 44$
$44 + 35 = 79$
Therefore,the next number in the series is $79$.

(B) The pattern followed in the series is $(\div 2) + 8$ for each step.
$4000 \div 2 = 2000; 2000 + 8 = 2008$
$2008 \div 2 = 1004; 1004 + 8 = 1012$
$1012 \div 2 = 506; 506 + 8 = 514$
$514 \div 2 = 257; 257 + 8 = 265$
$265 \div 2 = 132.5; 132.5 + 8 = 140.5$
$140.5 \div 2 = 70.25; 70.25 + 8 = 78.25$
Thus,the missing number is $514$.

(C) The pattern of the series is based on multiplying by consecutive odd numbers starting from $1$.
$5 \times 1 = 5$
$5 \times 3 = 15$
$15 \times 5 = 75$
$75 \times 7 = 525$
$525 \times 9 = 4725$
$4725 \times 11 = 51975$
Therefore,the missing number is $525$.

(A) The given series follows the pattern of multiplying by increasing factors of $0.5$:
$52 \times 0.5 = 26$
$26 \times 1 = 26$
$26 \times 1.5 = 39$
$39 \times 2 = 78$
$78 \times 2.5 = 195$
$195 \times 3 = 585$
Therefore,the missing number is $195$.

(B) Analyze the pattern of differences between consecutive terms:
$20 - 7 = 13$
$46 - 20 = 26$
$98 - 46 = 52$
$202 - 98 = 104$
The differences are $13, 26, 52, 104$,which follow a pattern of multiplying by $2$ ($13 \times 2 = 26$,$26 \times 2 = 52$,$52 \times 2 = 104$).
Following this pattern,the next difference should be $104 \times 2 = 208$.
Therefore,the next term is $202 + 208 = 410$.

(B) The pattern follows alternating operations of subtracting a cube and adding a square:
$210 - 1^3 = 210 - 1 = 209$
$209 + 2^2 = 209 + 4 = 213$
$213 - 3^3 = 213 - 27 = 186$
$186 + 4^2 = 186 + 16 = 202$
$202 - 5^3 = 202 - 125 = 77$
Thus,the next number in the series is $77$.

(D) The pattern in the series is based on adding multiples of $11$ with odd numbers:
$27 + (11 \times 1) = 27 + 11 = 38$
$38 + (11 \times 3) = 38 + 33 = 71$
$71 + (11 \times 5) = 71 + 55 = 126$
$126 + (11 \times 7) = 126 + 77 = 203$
Following this pattern,the next term is:
$203 + (11 \times 9) = 203 + 99 = 302$

(C) The given series is $435, 354, 282, 219, 165, ?$.
Let's analyze the differences between consecutive terms:
$435 - 354 = 81$
$354 - 282 = 72$
$282 - 219 = 63$
$219 - 165 = 54$
The differences are $81, 72, 63, 54$,which are multiples of $9$ in descending order $(9 \times 9, 9 \times 8, 9 \times 7, 9 \times 6)$.
The next difference should be $9 \times 5 = 45$.
Therefore,the missing number is $165 - 45 = 120$.

(C) The pattern follows the addition of consecutive squares of decreasing integers:
$4 + 14^2 = 4 + 196 = 200$
$200 + 13^2 = 200 + 169 = 369$
$369 + 12^2 = 369 + 144 = 513$
$513 + 11^2 = 513 + 121 = 634$
$634 + 10^2 = 634 + 100 = 734$
Therefore,the next number in the series is $734$.

(D) The given series is $325, 314, 288, 247, 191, ?$.
Let's find the difference between consecutive terms:
$325 - 314 = 11$
$314 - 288 = 26$
$288 - 247 = 41$
$247 - 191 = 56$
Now,observe the differences: $11, 26, 41, 56, ...$
The difference between these differences is constant:
$26 - 11 = 15$
$41 - 26 = 15$
$56 - 41 = 15$
So,the next difference should be $56 + 15 = 71$.
Therefore,the next term in the series is $191 - 71 = 120$.

(B) The pattern of the series is as follows:
$45 \times 1 + 1 = 46$
$46 \times 1.5 + 0 = 69$ (Wait,let's re-evaluate the pattern).
Let's check the differences:
$46 - 45 = 1$
$70 - 46 = 24$
$141 - 70 = 71$
This does not seem to be a simple difference series.
Let's try multiplication with factors:
$45 \times 0.5 + 23.5 = 46$
$46 \times 1 + 24 = 70$
$70 \times 1.5 + 36 = 141$
$141 \times 2 + 71.5 = 353.5$
$353.5 \times 2.5 + 177.75 = 1061.5$
Alternatively,checking the pattern: $45 \times 1 + 1 = 46$,$46 \times 1.5 + 0 = 69$ (No).
Let's re-examine: $45 \times 1 + 1 = 46$,$46 \times 1.5 + 0 = 69$ (No).
Correct pattern: $45 \times 0.5 + 23.5 = 46$,$46 \times 1 + 24 = 70$,$70 \times 1.5 + 36 = 141$,$141 \times 2 + 71.5 = 353.5$,$353.5 \times 2.5 + 177.75 = 1061.5$.
Actually,the simplest pattern is: $45 \times 1 + 1 = 46$,$46 \times 1.5 + 1 = 70$,$70 \times 2 + 1 = 141$,$141 \times 2.5 + 1 = 353.5$,$353.5 \times 3 + 1 = 1061.5$.

(C) The given series is $620, 632, 608, 644, 596, ?$.
Let's analyze the pattern between consecutive terms:
$620 + 12 = 632$
$632 - 24 = 608$
$608 + 36 = 644$
$644 - 48 = 596$
The pattern of differences is $+12, -24, +36, -48, ...$
Following this pattern,the next operation should be $+60$.
$596 + 60 = 656$.
Therefore,the missing number is $656$.

(C) The given series is $15, 25, 40, 65, ?, 170$.
Observe the pattern of the series:
$15 + 25 = 40$
$25 + 40 = 65$
$40 + 65 = 105$
$65 + 105 = 170$
Each term is the sum of the two preceding terms. Therefore,the missing number is $40 + 65 = 105$.

(B) The pattern in the series is as follows:
$9 \times 2 - 3 = 15$
$15 \times 2 - 3 = 27$
$27 \times 2 - 3 = 51$
$51 \times 2 - 3 = 99$
Following the same logic,the next number is:
$99 \times 2 - 3 = 198 - 3 = 195$
Therefore,the correct option is $B$.

(D) Analyze the differences between consecutive terms:
$21 - 13 = 8$
$36 - 21 = 15$
$58 - 36 = 22$
$87 - 58 = 29$
The differences are $8, 15, 22, 29$.
Observe the pattern in the differences:
$15 - 8 = 7$
$22 - 15 = 7$
$29 - 22 = 7$
The difference between consecutive differences is a constant $7$.
Therefore,the next difference should be $29 + 7 = 36$.
Adding this to the last term: $87 + 36 = 123$.

(D) The pattern of the series is based on adding consecutive odd squares plus $1$ to the previous term.
$7 + (1)^2 + 1 = 9$
$9 + (3)^2 + 1 = 19$
$19 + (5)^2 + 1 = 45$
$45 + (7)^2 + 1 = 95$
Following this pattern,the next term is:
$95 + (9)^2 + 1 = 95 + 81 + 1 = 177$

(A) Observe the pattern of differences between consecutive terms:
$15 - 14 = 1 = 1^2$
$23 - 15 = 8$ (This does not follow the square pattern directly).
Let us re-examine the series: $14, 15, 23, 32, 96, ?$
$14 + 1^3 = 15$
$15 + 2^3 = 23$
$23 + 3^2 = 32$
$32 + 4^2 = 48$ (Wait,the series provided is $14, 15, 23, 32, 96, ?$).
Actually,the pattern is: $14 + 1^2 = 15$,$15 + 2^3 = 23$,$23 + 3^2 = 32$,$32 + 4^3 = 96$,$96 + 5^2 = 121$.
The pattern alternates between adding the square and the cube of consecutive integers: $1^2, 2^3, 3^2, 4^3, 5^2$.
Therefore,the next term is $96 + 5^2 = 96 + 25 = 121$.

(C) Analyze the differences between consecutive terms:
$24 - 20 = 4$
$36 - 24 = 12$
$56 - 36 = 20$
$84 - 56 = 28$
The differences are $4, 12, 20, 28$,which form an arithmetic progression with a common difference of $8$.
The next difference should be $28 + 8 = 36$.
Therefore,the next term is $84 + 36 = 120$.

(A) The pattern in the series is as follows:
$4 \times 2 + 2 = 10$
$10 \times 3 + 10 = 40$
$40 \times 4 + 30 = 190$
$190 \times 5 + 190 = 940$
Alternatively,observing the differences:
$10 - 4 = 6$
$40 - 10 = 30$
$190 - 40 = 150$
$940 - 190 = 750$
The differences are $6, 30, 150, 750, \dots$ which follow the pattern $\times 5$.
Next difference $= 750 \times 5 = 3750$.
Missing number $= 940 + 3750 = 4690$.
Checking the next term: $4690 + (3750 \times 5) = 4690 + 18750 = 23440$. This matches the series.

(B) The pattern of the series is based on the difference between consecutive terms being halved.
$4000 - 2008 = 1992$; $1992 / 2 = 996$
$2008 - 1012 = 996$; $996 / 2 = 498$
$1012 - x = 498 \implies x = 1012 - 498 = 514$
$514 - 265 = 249$; $249 / 2 = 124.5$
$265 - 140.5 = 124.5$; $124.5 / 2 = 62.25$
$140.5 - 78.25 = 62.25$; $62.25 / 2 = 31.125$
Thus,the missing number is $514$.

(D) The pattern of the series is as follows:
$7 \times 0.5 + 0.5 = 4$
$4 \times 1 + 1 = 5$
$5 \times 1.5 + 1.5 = 9$
$9 \times 2 + 2 = 20$
$20 \times 2.5 + 2.5 = 52.5$
$52.5 \times 3 + 3 = 160.5$
Thus,the missing number is $20$.