Number Series Questions in English

(D) Let us analyze the differences between consecutive terms in the sequence: $7, 9, 17, 42, 91, 172, 293$.
$9 - 7 = 2$
$17 - 9 = 8$
$42 - 17 = 25$
$91 - 42 = 49$
$172 - 91 = 81$
$293 - 172 = 121$
The differences are $2, 8, 25, 49, 81, 121$. We observe that $49 = 7^2$,$81 = 9^2$,and $121 = 11^2$. The pattern of differences should be squares of odd numbers: $1^2, 3^2, 5^2, 7^2, 9^2, 11^2$,which are $1, 9, 25, 49, 81, 121$.
If the first term is $8$ instead of $7$,the sequence becomes $8, 9, 18, 43, 92, 173, 294$. However,looking at the original sequence $7, 9, 17, 42, 91, 172, 293$,the term $9$ is the odd one out because replacing it with $8$ makes the differences $1^2, 3^2, 5^2, 7^2, 9^2, 11^2$. Thus,$9$ is the incorrect term.

(B) Let us analyze the pattern of the sequence: $8, 14, 26, 48, 98, 194, 386$.
$14 = (8 \times 2) - 2 = 16 - 2 = 14$.
$26 = (14 \times 2) - 2 = 28 - 2 = 26$.
$50 = (26 \times 2) - 2 = 52 - 2 = 50$.
$98 = (50 \times 2) - 2 = 100 - 2 = 98$.
$194 = (98 \times 2) - 2 = 196 - 2 = 194$.
$386 = (194 \times 2) - 2 = 388 - 2 = 386$.
In the given sequence,$48$ does not follow the pattern $(x_{n} = 2x_{n-1} - 2)$. If we replace $48$ with $50$,the sequence becomes consistent. Thus,$48$ is the odd number in the sequence. Since the question asks to identify the odd number and provides $50$ as an option to replace it,option $(b)$ is the correct choice.

(B) Let us analyze the differences between consecutive terms in the sequence:
$95 - 86 = 9$
$86 - 73 = 13$
$73 - 62 = 11$
$62 - 47 = 15$
$47 - 30 = 17$
$30 - 11 = 19$
Observing the pattern of differences: $9, 13, 11, 15, 17, 19$.
If we replace $73$ with $75$,the sequence becomes $95, 86, 75, 62, 47, 30, 11$.
The new differences are:
$95 - 86 = 9$
$86 - 75 = 11$
$75 - 62 = 13$
$62 - 47 = 15$
$47 - 30 = 17$
$30 - 11 = 19$
The differences are now in the sequence $9, 11, 13, 15, 17, 19$,which follows a consistent pattern.
Thus,$73$ is the odd number in the sequence.
Therefore,option $(b)$ is the correct answer.

(C) Let us analyze the pattern of the given series:
$7 \times 2 = 14$
$14 \times 4 = 56$
$56 \times 3 = 168$
$168 \times 2 = 336$
$336 \times 4 = 1344$
$1344 \times 3 = 4032$
$4032 \times 2 = 8064$
The pattern follows the multiplication sequence of $\times 2, \times 4, \times 3, \times 2, \times 4, \times 3, \times 2$.
In the given sequence,the term $2688$ is incorrect and should be $4032$.
Therefore,the odd number in the sequence is $2688$,and the correct term that should be in its place is $4032$.
Hence,option $(c)$ is the correct alternative.

(B) The given sequence is $11, 15, 17, 19, 23, 25$.
Observing the prime numbers,we see that $11, 13, 17, 19, 23$ are consecutive prime numbers.
In the given sequence,$15$ and $25$ are composite numbers,while $11, 17, 19, 23$ are prime numbers.
However,if we look at the pattern of prime numbers starting from $11$,the sequence should be $11, 13, 17, 19, 23, 29$.
Among the given options,$15$ is the most distinct outlier as it breaks the sequence of consecutive prime numbers.
Therefore,$15$ is the odd number in the sequence.

(B) The pattern followed by the numbers of the given series is:
$9 = 8 \times 1 + 1$
$65 = 7 \times 9 + 2$
$393 = 6 \times 65 + 3$
Following the same logic for the second row:
$(a) = 8 \times 2 + 1 = 17$
$(b) = 7 \times 17 + 2 = 121$
$(c) = 6 \times 121 + 3 = 729$
$(d) = 5 \times 729 + 4 = 3649$
$(e) = 4 \times 3649 + 5 = 14601$
Therefore,the number in place of $(c)$ is $729$.

(A) The pattern in the first row is as follows:
$616 - 496 = 120 = 12 \times 10$
$496 - 397 = 99 = 11 \times 9$
$397 - 317 = 80 = 10 \times 8$
$317 - 254 = 63 = 9 \times 7$
Following the same pattern for the second row:
$(A) = 838 - 120 = 718$
$(B) = 718 - 99 = 619$
$(C) = 619 - 80 = 539$
$(D) = 539 - 63 = 476$
$(E) = 476 - (8 \times 6) = 476 - 48 = 428$

(D) The pattern follows the difference between consecutive terms in the top row:
$434 - 353 = 81 = 9^2$
$353 - 417 = -64 = -8^2$
$417 - 368 = 49 = 7^2$
$368 - 404 = -36 = -6^2$
$404 - 379 = 25 = 5^2$
Applying the same logic to the bottom row:
$108 - (A) = 9^2 = 81 \Rightarrow (A) = 108 - 81 = 27$
$27 - (B) = -8^2 = -64 \Rightarrow (B) = 27 + 64 = 91$
$91 - (C) = 7^2 = 49 \Rightarrow (C) = 91 - 49 = 42$
$42 - (D) = -6^2 = -36 \Rightarrow (D) = 42 + 36 = 78$
$78 - (E) = 5^2 = 25 \Rightarrow (E) = 78 - 25 = 53$
Thus,the value of $(E)$ is $53$.

(D) The pattern followed in the first row is: $x_{n} = 2 \times x_{n-1} + 8 \times (n-1)$,where $n$ is the position index starting from $2$.
For the second row,we start with $124$ and apply the same logic:
$(A) = 2 \times 124 + 8 \times 1 = 248 + 8 = 256$
$(B) = 2 \times 256 + 8 \times 2 = 512 + 16 = 528$
$(C) = 2 \times 528 + 8 \times 3 = 1056 + 24 = 1080$
$(D) = 2 \times 1080 + 8 \times 4 = 2160 + 32 = 2192$
$(E) = 2 \times 2192 + 8 \times 5 = 4384 + 40 = 4424$
Therefore,the value at $(C)$ is $1080$.

(B) The pattern followed by the numbers in the first row is:
$9 = (8 \times 1) + 1$
$65 = (7 \times 9) + 2$
$393 = (6 \times 65) + 3$
Applying the same pattern to the second row:
$(A) = (8 \times 2) + 1 = 17$
$(B) = (7 \times 17) + 2 = 119 + 2 = 121$
$(C) = (6 \times 121) + 3 = 726 + 3 = 729$
Thus,the number in place of $(C)$ is $729$.

(B) The pattern in the first row is: $x_{n+1} = \frac{x_n - 8}{2}$.
Checking the first row:
$420 = (848 - 8) / 2$
$206 = (420 - 8) / 2$
$99 = (206 - 8) / 2$
$45.5 = (99 - 8) / 2$
Applying the same logic to the second row starting with $664$:
$(A) = (664 - 8) / 2 = 328$
$(B) = (328 - 8) / 2 = 160$
$(C) = (160 - 8) / 2 = 76$
$(D) = (76 - 8) / 2 = 34$
Thus,the value of $(D)$ is $34$.

(C) The pattern in the first row is:
$8 \times 1 = 8$
$8 \times 1.5 = 12$
$12 \times 2 = 24$
$24 \times 2.5 = 60$
$60 \times 3 = 180$
Applying the same logic to the second row starting with $36$:
$(A) = 36 \times 1 = 36$
$(B) = 36 \times 1.5 = 54$
$(C) = 54 \times 2 = 108$
$(D) = 108 \times 2.5 = 270$
$(E) = 270 \times 3 = 810$

(D) Observe the pattern in the first row:
$14 = 1 \times 6 + 8$
$35 = 2 \times 14 + 7$
$111 = 3 \times 35 + 6$
$449 = 4 \times 111 + 5$
Following the same logic for the second row:
$(A) = 1 \times 3 + 8 = 11$
$(B) = 2 \times 11 + 7 = 29$
$(C) = 3 \times 29 + 6 = 93$
$(D) = 4 \times 93 + 5 = 377$
$(E) = 5 \times 377 + 4 = 1889$
Therefore,the value in place of $(B)$ is $29$.

(C) The pattern in the first row is:
$49 = (8 \times 7) - 7$
$288 = (49 \times 6) - 6$
$1435 = (288 \times 5) - 5$
$5736 = (1435 \times 4) - 4$
Applying the same logic to the second row starting with $5$:
$(A) = (5 \times 7) - 7 = 35 - 7 = 28$
$(B) = (28 \times 6) - 6 = 168 - 6 = 162$
$(C) = (162 \times 5) - 5 = 810 - 5 = 805$
$(D) = (805 \times 4) - 4 = 3220 - 4 = 3216$
$(E) = (3216 \times 3) - 3 = 9648 - 3 = 9645$

(C) The given series is $A = \frac{1}{0.4} + \frac{1}{0.04} + \frac{1}{0.004} + \dots$ up to $8$ terms.
We can rewrite the terms as:
$A = 2.5 + 25 + 250 + 2500 + \dots$ up to $8$ terms.
This is a Geometric Progression $(GP)$ where the first term $a = 2.5$ and the common ratio $r = 10$.
The sum of the first $n$ terms of a $GP$ is given by $S_n = \frac{a(r^n - 1)}{r - 1}$.
Here,$n = 8$,$a = 2.5$,and $r = 10$.
$A = \frac{2.5(10^8 - 1)}{10 - 1} = \frac{2.5(100000000 - 1)}{9} = \frac{2.5 \times 99999999}{9}$.
$A = 2.5 \times 11111111 = 27777777.5$.

(C) The first $111$ whole numbers are $0, 1, 2, \dots, 110$.
This is an arithmetic progression where the first term $a = 0$,the last term $l = 110$,and the number of terms $n = 111$.
The sum of an arithmetic progression is given by $S_n = \frac{n}{2}(a + l)$.
Substituting the values: $S_{111} = \frac{111}{2}(0 + 110)$.
$S_{111} = \frac{111}{2} \times 110$.
$S_{111} = 111 \times 55$.
$S_{111} = 6105$.
The unit digit of $6105$ is $5$.

(B) The series consists of $20$ terms,which can be grouped into two separate series,each having $10$ terms.
Let $S = S_1 + S_2$,where $S_1 = \sum_{n=1}^{10} \frac{1}{(2n-1)(2n+1)(2n+3)}$ and $S_2 = \sum_{n=1}^{10} \frac{1}{(3n-2)(3n+1)}$.
For $S_1$,the general term is $a_n = \frac{1}{(2n-1)(2n+1)(2n+3)} = \frac{1}{4} [\frac{1}{(2n-1)(2n+1)} - \frac{1}{(2n+1)(2n+3)}]$.
Summing $S_1$ from $n=1$ to $10$: $S_1 = \frac{1}{4} [(\frac{1}{1 \times 3} - \frac{1}{3 \times 5}) + (\frac{1}{3 \times 5} - \frac{1}{5 \times 7}) + \dots + (\frac{1}{19 \times 21} - \frac{1}{21 \times 23})] = \frac{1}{4} [\frac{1}{3} - \frac{1}{483}] = \frac{1}{4} [\frac{161-1}{483}] = \frac{160}{4 \times 483} = \frac{40}{483}$.
For $S_2$,the general term is $b_n = \frac{1}{(3n-2)(3n+1)} = \frac{1}{3} [\frac{1}{3n-2} - \frac{1}{3n+1}]$.
Summing $S_2$ from $n=1$ to $10$: $S_2 = \frac{1}{3} [(1 - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{7}) + \dots + (\frac{1}{28} - \frac{1}{31})] = \frac{1}{3} [1 - \frac{1}{31}] = \frac{1}{3} [\frac{30}{31}] = \frac{10}{31}$.
Total sum $S = S_1 + S_2 = \frac{40}{483} + \frac{10}{31} = \frac{40 \times 31 + 10 \times 483}{483 \times 31} = \frac{1240 + 4830}{14973} = \frac{6070}{14973}$.

(D) The given series is $14^{3} + 16^{3} + 18^{3} + \ldots + 30^{3}$.
This can be written as $\sum_{k=7}^{15} (2k)^{3} = 8 \sum_{k=7}^{15} k^{3}$.
Using the formula for the sum of cubes,$\sum_{k=1}^{n} k^{3} = [\frac{n(n+1)}{2}]^{2}$.
We need to calculate $8 [(\sum_{k=1}^{15} k^{3}) - (\sum_{k=1}^{6} k^{3})]$.
$\sum_{k=1}^{15} k^{3} = [\frac{15 \times 16}{2}]^{2} = (120)^{2} = 14400$.
$\sum_{k=1}^{6} k^{3} = [\frac{6 \times 7}{2}]^{2} = (21)^{2} = 441$.
Sum $= 8 \times (14400 - 441) = 8 \times 13959 = 111672$.

(NONE) Given:
First term $(a)$ = $20$
Last term $(l)$ = $28$
Number of terms $(n)$ = $17$
The formula for the sum of an arithmetic progression when the first and last terms are known is:
$S_n = \frac{n}{2} (a + l)$
Substituting the values:
$S_{17} = \frac{17}{2} (20 + 28)$
$S_{17} = \frac{17}{2} (48)$
$S_{17} = 17 \times 24$
$S_{17} = 408$

(D) Given:
First term $(a)$ = $7$
Last term $(l)$ = $55$
Number of terms $(n)$ = $9$
The formula for the sum of the first $n$ terms of an arithmetic progression when the first and last terms are known is:
$S_n = \frac{n}{2}(a + l)$
Substituting the given values:
$S_9 = \frac{9}{2}(7 + 55)$
$S_9 = \frac{9}{2}(62)$
$S_9 = 9 \times 31$
$S_9 = 279$

(A) Given: The first term $a = -10$,the last term $l = 26$,and the number of terms $n = 13$.
The formula for the sum of the first $n$ terms of an arithmetic progression when the first and last terms are known is $S_n = \frac{n}{2}(a + l)$.
Substituting the given values into the formula:
$S_{13} = \frac{13}{2}(-10 + 26)$
$S_{13} = \frac{13}{2}(16)$
$S_{13} = 13 \times 8$
$S_{13} = 104$.
Therefore,the sum of the first $13$ terms is $104$.

(C) Let the first term be $a$ and the common difference be $d$. The $n^{th}$ term of an arithmetic progression is given by $t_n = a + (n-1)d$.
Given:
$t_3 = a + 2d = -9$ --- (Equation $1$)
$t_7 = a + 6d = 11$ --- (Equation $2$)
Subtracting Equation $1$ from Equation $2$:
$(a + 6d) - (a + 2d) = 11 - (-9)$
$4d = 20$
$d = 5$
Substituting $d = 5$ into Equation $1$:
$a + 2(5) = -9$
$a + 10 = -9$
$a = -19$
Now,finding the $15^{th}$ term $(t_{15})$:
$t_{15} = a + 14d$
$t_{15} = -19 + 14(5)$
$t_{15} = -19 + 70$
$t_{15} = 51$

(A) The given series is $3, 7, 16, 35, 70, 153$.
Let's analyze the pattern:
$3 \times 2 + 1 = 7$
$7 \times 2 + 2 = 16$
$16 \times 2 + 3 = 35$
$35 \times 2 + 4 = 74$
$74 \times 2 + 5 = 153$
Comparing this with the given series,the term $70$ is incorrect because the pattern requires $74$ at that position.
Therefore,the wrong number is $70$.

(D) The given series is an alternating series consisting of two interleaved sequences.
Sequence $1$: $123, 106, 89, \dots$
Pattern: $123 - 17 = 106$,$106 - 17 = 89$.
Sequence $2$: $140, 157, ?$
Pattern: $140 + 17 = 157$,$157 + 17 = 174$.
Therefore,the missing number is $174$.

(C) The pattern in the given series is that each term is obtained by dividing the previous term by $2$ and then subtracting $1$.
$190 \div 2 - 1 = 95 - 1 = 94$
$94 \div 2 - 1 = 47 - 1 = 46$
$46 \div 2 - 1 = 23 - 1 = 22$
$22 \div 2 - 1 = 11 - 1 = 10$
To verify,$10 \div 2 - 1 = 5 - 1 = 4$,which matches the last term.
Therefore,the missing number is $10$.

(B) The pattern follows the subtraction of $6 \times n$ where $n$ is a sequence of differences.
$320 - 6 \times 0 = 320$
$320 - 6 \times 1 = 314$
$314 - 6 \times 4 = 290$
$290 - 6 \times 10 = 230$
The sequence of multipliers is $0, 1, 4, 10, 20$. The differences between these multipliers are $1, 3, 6, 10$,which are triangular numbers.
Next term in the sequence of multipliers is $10 + 10 = 20$.
Therefore,the next term is $230 - 6 \times 20 = 230 - 120 = 110$.

(D) The pattern followed in the series is as follows:
$3 \times 1 + 1 = 4$
$4 \times 2 + 1 = 9$
$9 \times 3 + 1 = 28$
$28 \times 4 + 1 = 113$
Following the same logic,the next term is:
$113 \times 5 + 1 = 565 + 1 = 566$
Thus,the missing number is $566$.

(C) The pattern follows the multiplication of consecutive odd numbers divided by $2$:
$8 \times 0.5 = 4$
$4 \times 1.5 = 6$
$6 \times 2.5 = 15$
$15 \times 3.5 = 52.5$
$52.5 \times 4.5 = 236.25$
Thus,the missing number is $52.5$.

(D) The given series is $3, 5, 13, 49, 241, ?$.
Observe the pattern between consecutive terms:
$3 \times 2 - 1 = 5$
$5 \times 3 - 2 = 13$
$13 \times 4 - 3 = 49$
$49 \times 5 - 4 = 241$
The pattern follows the rule: $(\text{Previous term} \times n) - (n - 1)$,where $n$ starts from $2$ and increases by $1$ for each step.
Following this pattern for the next term:
$241 \times 6 - 5 = 1446 - 5 = 1441$.
Therefore,the missing number is $1441$.

(C) The given series is $7, 13, 31, 85, 247, ?$.
Observe the pattern between consecutive terms:
$7 \times 3 - 8 = 13$
$13 \times 3 - 8 = 31$
$31 \times 3 - 8 = 85$
$85 \times 3 - 8 = 247$
Following the same pattern,the next term is:
$247 \times 3 - 8 = 741 - 8 = 733$
Thus,the missing number is $733$.

(B) Let the given series be $5, 7, 17, 47, 115, x$.
Calculate the differences between consecutive terms:
$7 - 5 = 2$
$17 - 7 = 10$
$47 - 17 = 30$
$115 - 47 = 68$
Now,find the second level of differences:
$10 - 2 = 8$
$30 - 10 = 20$
$68 - 30 = 38$
Now,find the third level of differences:
$20 - 8 = 12$
$38 - 20 = 18$
Observe the pattern in the third level of differences: $12, 18$. The difference increases by $6$ each time. So,the next difference in the third level should be $18 + 6 = 24$.
Now,work backwards to find $x$:
Next second level difference $= 38 + 24 = 62$.
Next first level difference $= 68 + 62 = 130$.
Therefore,$x = 115 + 130 = 245$.

(B) The given series is $508, 256, 130, 67, 35.5, ?$.
Observe the pattern:
$508 \div 2 + 2 = 254 + 2 = 256$
$256 \div 2 + 2 = 128 + 2 = 130$
$130 \div 2 + 2 = 65 + 2 = 67$
$67 \div 2 + 2 = 33.5 + 2 = 35.5$
Following the same logic for the next term:
$35.5 \div 2 + 2 = 17.75 + 2 = 19.75$
Therefore,the missing number is $19.75$.

(C) The pattern follows the rule: $(n + 1) \times \frac{k}{2}$,where $k$ is an increasing odd number sequence $(1, 3, 5, 7, 9, \dots)$.
Step $1$: $(17 + 1) \times \frac{1}{2} = 18 \times 0.5 = 9$
Step $2$: $(9 + 1) \times \frac{3}{2} = 10 \times 1.5 = 15$
Step $3$: $(15 + 1) \times \frac{5}{2} = 16 \times 2.5 = 40$
Step $4$: $(40 + 1) \times \frac{7}{2} = 41 \times 3.5 = 143.5$
Step $5$: $(143.5 + 1) \times \frac{9}{2} = 144.5 \times 4.5 = 650.25$

(B) The given sequence is $40960, 10240, 2560, 640, 200, 40, 10$.
Observe the pattern between consecutive terms:
$40960 \div 4 = 10240$
$10240 \div 4 = 2560$
$2560 \div 4 = 640$
$640 \div 4 = 160$
Comparing this with the given sequence,the term $200$ is incorrect,as the expected value is $160$.
Continuing the pattern with $160$:
$160 \div 4 = 40$
$40 \div 4 = 10$
Therefore,the wrong number in the sequence is $200$.

(A) Let us analyze the differences between consecutive terms in the sequence:
$45 - 41 = 4 = 2^2$
$61 - 45 = 16 = 4^2$
$97 - 61 = 36 = 6^2$
$181 - 97 = 84$ (This is not $8^2 = 64$)
$261 - 181 = 80$
$405 - 261 = 144 = 12^2$
If we follow the pattern of adding squares of even numbers $(2^2, 4^2, 6^2, 8^2, 10^2, 12^2)$:
$41 + 4 = 45$
$45 + 16 = 61$
$61 + 36 = 97$
$97 + 64 = 161$
$161 + 100 = 261$
$261 + 144 = 405$
Comparing this with the given sequence,$181$ is the incorrect term and should be $161$.

(D) Let us analyze the differences between consecutive terms in the sequence:
$30 - 16 = 14$
$58 - 30 = 28$
$114 - 58 = 56$
$226 - 114 = 112$
$496 - 226 = 270$
$898 - 496 = 402$
The pattern of differences is $14, 28, 56, 112, 224, 448, \dots$ (each difference is double the previous one).
If we follow this pattern starting from $226$,the next difference should be $224$.
$226 + 224 = 450$.
Then,the next difference should be $448$.
$450 + 448 = 898$.
Since $898$ is correct,the term $496$ is the odd one out,as it should be $450$.

(D) The pattern of the series is $\times 1 + 5.5, \times 2 + 5.5, \times 3 + 5.5, \times 4 + 5.5, \times 5 + 5.5, \times 6 + 5.5$.
Calculating the terms:
$15 \times 1 + 5.5 = 20.5$
$20.5 \times 2 + 5.5 = 46.5$
$46.5 \times 3 + 5.5 = 145$
$145 \times 4 + 5.5 = 585.5$
$585.5 \times 5 + 5.5 = 2933$
$2933 \times 6 + 5.5 = 17603.5$
Comparing this with the given sequence,the term $21.5$ is incorrect and should be $20.5$.

(D) The pattern of the series is $\times 1 + 1^2, \times 2 + 2^2, \times 3 + 3^2, \times 4 + 4^2, \times 5 + 5^2, \times 6 + 6^2$.
Step-by-step calculation:
$5 \times 1 + 1^2 = 5 + 1 = 6$
$6 \times 2 + 2^2 = 12 + 4 = 16$
$16 \times 3 + 3^2 = 48 + 9 = 57$
$57 \times 4 + 4^2 = 228 + 16 = 244$
$244 \times 5 + 5^2 = 1220 + 25 = 1245$
$1245 \times 6 + 6^2 = 7470 + 36 = 7506$
Comparing this with the given sequence,$246$ is incorrect and should be $244$.

(B) The pattern of the series is based on adding multiples of $3$ to the previous difference:
$13 - 2 = 11$
$46 - 13 = 33$ $(11 \times 3)$
$145 - 46 = 99$ $(33 \times 3)$
$442 - 145 = 297$ $(99 \times 3)$
$1333 - 442 = 891$ $(297 \times 3)$
$4006 - 1333 = 2673$ $(891 \times 3)$
Comparing this with the given sequence,$452$ is incorrect because the difference $452 - 145 = 307$,which does not follow the pattern. The correct term should be $442$.

(A) The given sequence follows the pattern $n^{3} - 1$,where $n$ is a natural number starting from $1$.
For $n=1$: $1^{3} - 1 = 1 - 1 = 0$
For $n=2$: $2^{3} - 1 = 8 - 1 = 7$
For $n=3$: $3^{3} - 1 = 27 - 1 = 26$
For $n=4$: $4^{3} - 1 = 64 - 1 = 63$
For $n=5$: $5^{3} - 1 = 125 - 1 = 124$
For $n=6$: $6^{3} - 1 = 216 - 1 = 215$
Comparing this with the given sequence,the last term is $217$,but according to the pattern,it should be $215$. Therefore,$217$ is the odd number in the sequence.

(A) The pattern in the series is as follows:
$7 \times 1 + 1 = 8$
$8 \times 2 + 2 = 18$
$18 \times 3 + 3 = 57$
Following the same logic,the next term is:
$57 \times 4 + 4 = 228 + 4 = 232$
Therefore,the missing number is $232$.

(D) The pattern in the series is as follows:
$7 + 4 = 11$
$11 + 8 = 19$
$19 + 16 = 35$
$35 + 32 = 67$
The differences between consecutive terms are powers of $2$ $(4, 8, 16, 32)$.
Thus,the next term is $35 + 32 = 67$.

(C) The given series follows the pattern: $x_{n+1} = (x_n \times 2) + 1$.
Step $1$: $5 \times 2 + 1 = 11$.
Step $2$: $11 \times 2 + 1 = 23$.
Step $3$: $23 \times 2 + 1 = 47$.
Step $4$: $47 \times 2 + 1 = 95$.
Therefore,the missing number is $47$.

(D) To find the next number in the series $17, 22, 52, 165, ?$,we analyze the pattern between consecutive terms:
$17 \times 1 + 5 = 22$
$22 \times 2 + 8 = 52$
$52 \times 3 + 9 = 165$
$165 \times 4 + 8 = 668$
Alternatively,looking at the pattern of the additions $(5, 8, 9, 8)$:
The multipliers are $1, 2, 3, 4$.
The additions follow a pattern where the second difference is constant $(-2)$:
$5 \xrightarrow{+3} 8 \xrightarrow{+1} 9 \xrightarrow{-1} 8$
Differences: $3, 1, -1$. The difference between these is $-2$.
Following this logic,the next term is $165 \times 4 + 8 = 660 + 8 = 668$.

(A) The pattern of the series is based on multiplying by consecutive prime numbers:
$2 \times 3 = 6$
$6 \times 5 = 30$
$30 \times 7 = 210$
$210 \times 11 = 2310$
$2310 \times 13 = 30030$
Thus,the missing term $x$ is $2310$.

(D) The given series follows a specific alternating pattern of multiplication and subtraction:
$3 \times 6 = 18$
$18 - 6 = 12$
$12 \times 6 = 72$
$72 - 6 = 66$
$66 \times 6 = 396$
Following this pattern,the next step is to subtract $6$ from the last number:
$396 - 6 = 390$
Therefore,the missing number is $390$.

(A) Let us analyze the differences between consecutive terms in the sequence:
$5555 - 5506 = 49 = 7^2$
$5506 - 5425 = 81 = 9^2$
$5425 - 5344 = 81$ (This pattern seems inconsistent).
Let us re-examine the sequence: $5555, 5506, 5425, 5344, 5135, 4910, 4621$.
Differences:
$5555 - 5506 = 49 = 7^2$
$5506 - 5425 = 81 = 9^2$
$5425 - 5304 = 121 = 11^2$
$5304 - 5135 = 169 = 13^2$
$5135 - 4910 = 225 = 15^2$
$4910 - 4621 = 289 = 17^2$
Comparing the original sequence $5531, 5506, 5425, 5344, 5135, 4910, 4621$ with the pattern,the term $5531$ is incorrect and should be $5555$,and $5344$ is the correct term instead of $534$. Thus,$5531$ is the odd one out.

(B) The given sequence is $6, 7, 9, 13, 26, 37, 69$.
Let's analyze the differences between consecutive terms:
$7 - 6 = 1$
$9 - 7 = 2$
$13 - 9 = 4$
$21 - 13 = 8$
$37 - 21 = 16$
$69 - 37 = 32$
The pattern of differences is $1, 2, 4, 8, 16, 32$,which corresponds to powers of $2$ $(2^0, 2^1, 2^2, 2^3, 2^4, 2^5)$.
In the original sequence,$26$ is present instead of $21$. Therefore,$26$ is the odd number in the sequence.

(D) Let us analyze the pattern in the sequence:
$1 \times 1 + 2 = 3$
$3 \times 2 + 4 = 10$
$10 \times 3 + 6 = 36$
$36 \times 4 + 8 = 152$
$152 \times 5 + 10 = 770$
$770 \times 6 + 12 = 4632$
Comparing this with the given sequence,the number $760$ is incorrect and should be $770$.

(D) Let us analyze the differences between consecutive terms:
$3 - 4 = -1$
$9 - 3 = 6$
$34 - 9 = 25$
$96 - 34 = 62$
$219 - 96 = 123$
$435 - 219 = 216$
Now,let us look at the differences of the differences:
$6 - (-1) = 7$
$25 - 6 = 19$
$62 - 25 = 37$
$123 - 62 = 61$
$216 - 123 = 93$
Looking at the second differences: $7, 19, 37, 61, 93$. The third differences are:
$19 - 7 = 12$
$37 - 19 = 18$
$61 - 37 = 24$
$93 - 61 = 32$
The third differences are $12, 18, 24, 32$. The pattern should be $12, 18, 24, 30$. If the last difference was $30$ instead of $32$,then the second difference would be $61 + 30 = 91$,and the first difference would be $123 + 91 = 214$. Thus,$219 + 214 = 433$. Therefore,$435$ is the odd number.