Number Series Questions in English

(C) The pattern in the series is as follows:
$1 + 3 = 4$
$4 + 6 = 10$
$10 + 12 = 22$
$22 + 24 = 46$
In this series,the difference between consecutive terms doubles each time $(3, 6, 12, 24, ...)$.
Therefore,the next difference should be $24 \times 2 = 48$.
Adding this to the last term: $46 + 48 = 94$.

(C) Analyze the difference between consecutive terms:
$0.55 - 0.5 = 0.05$
$0.65 - 0.55 = 0.10$
$0.8 - 0.65 = 0.15$
The differences are increasing by $0.05$ each time $(0.05, 0.10, 0.15, ...)$.
Therefore,the next difference should be $0.15 + 0.05 = 0.20$.
Adding this to the last term: $0.8 + 0.20 = 1.0$.

(B) The given series is $5, 6, 9, 15, ?, 40$.
Let's analyze the differences between consecutive terms:
$6 - 5 = 1$
$9 - 6 = 3$
$15 - 9 = 6$
The differences are $1, 3, 6, ...$
Now,let's look at the differences of these differences:
$3 - 1 = 2$
$6 - 3 = 3$
This shows that the second-level differences are increasing by $1$ $(2, 3, 4, 5, ...)$.
Following this pattern,the next difference in the first level should be $6 + 4 = 10$.
Therefore,the missing number is $15 + 10 = 25$.
To verify,the next difference should be $10 + 5 = 15$,and $25 + 15 = 40$,which matches the last term in the series.
Thus,the missing number is $25$.

(C) The given series is a sequence of squares of consecutive odd numbers:
$1^{2} = 1$
$3^{2} = 9$
$5^{2} = 25$
$7^{2} = 49$
$9^{2} = 81$
Following this pattern,the next odd number is $11$,and its square is $11^{2} = 121$.

(B) The given sequence is a combination of two alternating series.
The first series consists of the squares of natural numbers: $1^2, 2^2, 3^2, 4^2, \dots$ which are $1, 4, 9, 16$.
The second series consists of the cubes of natural numbers: $1^3, 2^3, 3^3, ?^3$ which are $1, 8, 27, ?$.
The next term in the sequence corresponds to the fourth term of the second series,which is $4^3 = 64$.

(D) The given series is a geometric progression where each term is obtained by multiplying the previous term by $3$.
$4 \times 3 = 12$
$12 \times 3 = 36$
$36 \times 3 = 108$
$108 \times 3 = 324$
Therefore,the next term in the series is $324$.

(D) The given series follows the pattern of multiplying the previous term by consecutive integers starting from $1$.
$1 \times 1 = 1$
$1 \times 2 = 2$
$2 \times 3 = 6$
$6 \times 4 = 24$
$24 \times 5 = 120$
$120 \times 6 = 720$
Therefore,the missing number is $120$.

(B) Observe the pattern of division in the series:
$120 \div 1 = 120$
$120 \div 2 = 60$
$60 \div 3 = 20$
$20 \div 4 = 5$
$5 \div 5 = 1$
Following this pattern,the missing number is $120 \div 1 = 120$.

(A) The given series is $4, 6, 9, ?$.
Observe the pattern of differences between consecutive terms:
$6 - 4 = 2$
$9 - 6 = 3$
The differences are increasing by $1$ $(2, 3, 4, ...)$.
Following this pattern,the next difference should be $4$.
Therefore,the next term is $9 + 4 = 13$.

(C) The pattern of the series is based on division by decreasing integers starting from $6$.
$5760 \div 6 = 960$
$960 \div 5 = 192$
$192 \div 4 = 48$
$48 \div 3 = 16$
$16 \div 2 = 8$
Therefore,the missing number is $192$.

(C) The pattern in the series is as follows:
$1 + 1 = 2$
$2 \times 3 = 6$
$6 + 1 = 7$
$7 \times 3 = 21$
$21 + 1 = 22$
$22 \times 3 = 66$
$66 + 1 = 67$
Following this alternating pattern of adding $1$ and multiplying by $3$,the next step is to multiply by $3$:
$67 \times 3 = 201$
Therefore,the missing number is $201$.

(C) The given series follows a repeating pattern of operations: $\div 2$ and $\times 4$.
$48 \div 2 = 24$
$24 \times 4 = 96$
$96 \div 2 = 48$
$48 \times 4 = 192$
Following this pattern,the next step is $192 \div 2 = 96$.

(B) The given sequence is a combination of two alternating series.
First series (odd positions): $1, 3, 9, ?$
In this series,each term is multiplied by $3$ to get the next term ($1 \times 3 = 3$,$3 \times 3 = 9$,$9 \times 3 = 27$).
Second series (even positions): $2, 6, 18, 54$
In this series,each term is also multiplied by $3$ ($2 \times 3 = 6$,$6 \times 3 = 18$,$18 \times 3 = 54$).
Therefore,the missing term is $9 \times 3 = 27$.

(A) The pattern in the given series is an alternating addition of $30$ and $60$.
$165 + 30 = 195$
$195 + 60 = 255$
$255 + 30 = 285$
$285 + 60 = 345$
Following this pattern,the next step is to add $30$ to the last term:
$345 + 30 = 375$
Therefore,the missing number is $375$.

(B) Observe the pattern in the given series:
$9 \times 3 = 27$
$27 + 4 = 31$
$31 \times 5 = 155$
$155 + 6 = 161$
$161 \times 7 = 1127$
Following the alternating pattern of multiplication and addition with increasing integers $(3, 4, 5, 6, 7, 8)$,the next step is to add $8$ to the last number:
$1127 + 8 = 1135$

(C) The pattern in the series is as follows:
$2 + 1 = 3$
$3 \times 1 = 3$
$3 + 2 = 5$
$5 \times 2 = 10$
$10 + 3 = 13$
$13 \times 3 = 39$
$39 + 4 = 43$
$43 \times 4 = 172$
$172 + 5 = 177$
Thus,the missing number is $39$.

(B) The given series follows the pattern of $(n^{2} - 1)$ for even numbers starting from $n = 2$.
$2^{2} - 1 = 4 - 1 = 3$
$4^{2} - 1 = 16 - 1 = 15$
$6^{2} - 1 = 36 - 1 = 35$
$8^{2} - 1 = 64 - 1 = 63$
$10^{2} - 1 = 100 - 1 = 99$
$12^{2} - 1 = 144 - 1 = 143$
Therefore,the missing term is $6^{2} - 1 = 35$.

(D) The given series follows the pattern of $(n^{3} - 1)$ for $n = 2, 3, 4, 5, 6, 7, ...$
$2^{3} - 1 = 8 - 1 = 7$
$3^{3} - 1 = 27 - 1 = 26$
$4^{3} - 1 = 64 - 1 = 63$
$5^{3} - 1 = 125 - 1 = 124$
$6^{3} - 1 = 216 - 1 = 215$
$7^{3} - 1 = 343 - 1 = 342$
Following this pattern,the next term is $8^{3} - 1 = 512 - 1 = 511$.

(B) The given series is $3, 7, 15, ?, 63, 127$.
Observe the pattern between consecutive terms:
$3 \times 2 + 1 = 7$
$7 \times 2 + 1 = 15$
Following this pattern,the next term is $15 \times 2 + 1 = 31$.
To verify,check the subsequent terms:
$31 \times 2 + 1 = 63$
$63 \times 2 + 1 = 127$
The pattern holds true.
Therefore,the missing term is $31$.

(B) The given series follows the pattern: $x_{n+1} = x_n \times 3 - 2$.
Step $1$: $4 \times 3 - 2 = 12 - 2 = 10$
Step $2$: $10 \times 3 - 2 = 30 - 2 = 28$
Step $3$: $28 \times 3 - 2 = 84 - 2 = 82$
Step $4$: $82 \times 3 - 2 = 246 - 2 = 244$
Step $5$: $244 \times 3 - 2 = 732 - 2 = 730$
Thus,the missing term is $28$.

(A) The given series is $3, 12, 27, 48, 75, 108, ?$.
Let's find the difference between consecutive terms:
$12 - 3 = 9$
$27 - 12 = 15$
$48 - 27 = 21$
$75 - 48 = 27$
$108 - 75 = 33$
The differences are $9, 15, 21, 27, 33$.
Now,find the difference between these differences:
$15 - 9 = 6$
$21 - 15 = 6$
$27 - 21 = 6$
$33 - 27 = 6$
Since the second difference is constant $(6)$,the next difference in the first series will be $33 + 6 = 39$.
Therefore,the next term in the series is $108 + 39 = 147$.

(D) The pattern follows an alternating addition and subtraction of multiples of $84$ by powers of $2$.
$563 + 84 \times 2^0 = 563 + 84 = 647$
$647 - 84 \times 2^1 = 647 - 168 = 479$
$479 + 84 \times 2^2 = 479 + 336 = 815$
$815 - 84 \times 2^3 = 815 - 672 = 143$
Therefore,the next term in the series is $143$.

(A) The given series is $5, 2, 7, 9, 16, 25, ?$.
Observe the pattern:
$5 + 2 = 7$
$2 + 7 = 9$
$7 + 9 = 16$
$9 + 16 = 25$
This is a Fibonacci-like series where each term is the sum of the two preceding terms.
Therefore,the next term is $16 + 25 = 41$.

(D) The pattern follows the rule: the next term is the sum of the previous two terms plus $2$.
$10 + 14 + 2 = 26$
$14 + 26 + 2 = 42$
$26 + 42 + 2 = 70$
$42 + 70 + 2 = 114$
Therefore,the missing number is $114$.

(C) The pattern in the series is that each term is the product of the two preceding terms.
$2 \times 8 = 16$
$8 \times 16 = 128$
$16 \times 128 = 2048$
Therefore,the next term in the series is $2048$.

(C) The pattern follows the rule $x_{n+1} = (x_n)^2 + 1$.
For the first term: $3^2 + 1 = 9 + 1 = 10$.
For the second term: $10^2 + 1 = 100 + 1 = 101$.
Following this pattern,the next term is: $101^2 + 1 = 10201 + 1 = 10202$.

(D) The pattern follows the removal of one digit from the left and right ends alternatively.
$1$. Start: $589654237$
$2$. Remove left digit $(5)$: $89654237$
$3$. Remove right digit $(7)$: $8965423$
$4$. Remove left digit $(8)$: $965423$
$5$. Remove right digit $(3)$: $96542$
Therefore,the next number in the series is $96542$.

(B) The pattern follows the rule: $\text{Next term} = \text{Previous term} - \text{first } 3 \text{ digits of the previous term}$.
$5824 - 582 = 5242$
$5242 - 524 = 4718$
$4718 - 471 = 4247$
$4247 - 424 = 3823$
Thus,the missing term is $4718$.

(C) The pattern follows that each term is the sum of the three preceding terms,starting from the fourth term $(8)$:
$1 + 3 + 4 = 8$
$3 + 4 + 8 = 15$
$4 + 8 + 15 = 27$
Following this logic,the next term is the sum of the three terms preceding it:
$8 + 15 + 27 = 50$
Therefore,the next term in the series is $50$.

(C) The pattern follows the multiplication of the digits of the previous number to obtain the next number.
$66 \rightarrow 6 \times 6 = 36$
$36 \rightarrow 3 \times 6 = 18$
$18 \rightarrow 1 \times 8 = 8$
Therefore,the next number in the series is $8$.

(B) The given sequence is a combination of $3$ alternating series.
$1^{\text{st}}$ series: $0, 3, 6, \dots$ (each term increases by $3$)
$2^{\text{nd}}$ series: $4, 7, ?$
$3^{\text{rd}}$ series: $6, 9, 12, \dots$ (each term increases by $3$)
In the $2^{\text{nd}}$ series,each term is obtained by adding $3$ to the previous term. Therefore,the missing term is $7 + 3 = 10$.

(A) The given series is a combination of $3$ alternating series.
$1^{\text{st}}$ series: $8, 7, 6, ?$
In this series,each term is obtained by subtracting $1$ from the previous term $(8-1=7, 7-1=6, 6-1=5)$.
$2^{\text{nd}}$ series: $9, 10, 11, 12$
In this series,each term is obtained by adding $1$ to the previous term $(9+1=10, 10+1=11, 11+1=12)$.
$3^{\text{rd}}$ series: $8, 9, 10, \dots$
In this series,each term is obtained by adding $1$ to the previous term $(8+1=9, 9+1=10)$.
Following the pattern of the $1^{\text{st}}$ series,the missing term is $6-1=5$.

(C) 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 term is $20$.

(C) The series follows the pattern of adding cubes of consecutive decreasing integers:
$5 + 7^3 = 5 + 343 = 348$
$348 + 6^3 = 348 + 216 = 564$
$564 + 5^3 = 564 + 125 = 689$
$689 + 4^3 = 689 + 64 = 753$
$753 + 3^3 = 753 + 27 = 780$
$780 + 2^3 = 780 + 8 = 788$
Thus,the missing number is $753$.

(B) Observe the differences between consecutive terms in the series:
$16 - 4.5 = 11.5$
$33 - 25.5 = 7.5$
$38.5 - 33 = 5.5$
$42 - 38.5 = 3.5$
$43.5 - 42 = 1.5$
The differences are decreasing by $2$ at each step: $11.5, 9.5, 7.5, 5.5, 3.5, 1.5$.
Therefore,the missing term is $16 + 9.5 = 25.5$ and $25.5 + 7.5 = 33$.
Thus,the missing term is $25.5$.

(A) The pattern follows the addition of squares and cubes of consecutive even numbers in an alternating sequence:
$14 + 2^2 = 14 + 4 = 18$
$18 + 4^3 = 18 + 64 = 82$
$82 + 6^2 = 82 + 36 = 118$
$118 + 8^3 = 118 + 512 = 630$
$630 + 10^2 = 630 + 100 = 730$
Therefore,the next number is $730$.

(C) The pattern follows the logic: $(5n) \times (n+1)^2 \pm (15 - n + 1)$ where $n$ starts from $1$.
For $n=1$: $5 \times 2^2 + 15 = 35$
For $n=2$: $10 \times 3^2 - 14 = 76$
For $n=3$: $15 \times 4^2 + 13 = 253$
For $n=4$: $20 \times 5^2 - 12 = 488$
For $n=5$: $25 \times 6^2 + 11 = 911$
For $n=6$: $30 \times 7^2 - 10 = 1460$
Thus,the next term in the series is $1460$.

(B) The pattern in the given series is based on adding multiples of $13$ in decreasing order:
$19 + (13 \times 6) = 19 + 78 = 97$
$97 + (13 \times 5) = 97 + 65 = 162$
$162 + (13 \times 4) = 162 + 52 = 214$
$214 + (13 \times 3) = 214 + 39 = 253$
Following this pattern,the next term is:
$253 + (13 \times 2) = 253 + 26 = 279$

(C) Analyze the difference between consecutive terms:
$15 - 9 = 6$
$26 - 15 = 11$
$42 - 26 = 16$
$63 - 42 = 21$
The differences are $6, 11, 16, 21$,which form an arithmetic progression with a common difference of $5$.
The next difference should be $21 + 5 = 26$.
Therefore,the next term is $63 + 26 = 89$.

(C) The pattern follows the addition of squares and cubes of consecutive integers starting from $2$:
$6 + 2^2 = 6 + 4 = 10$
$10 + 3^3 = 10 + 27 = 37$
$37 + 4^2 = 37 + 16 = 53$
$53 + 5^3 = 53 + 125 = 178$
$178 + 6^2 = 178 + 36 = 214$
Therefore,the next number in the series is $214$.

(C) The given series follows the pattern: $T_{n+1} = T_n \times (6-n) - (7-n) \times (6-n)$ for $n=0, 1, 2, 3, 4$.
Specifically:
$30 = 12 \times 6 - 7 \times 6 = 72 - 42 = 30$
$120 = 30 \times 5 - 6 \times 5 = 150 - 30 = 120$
$460 = 120 \times 4 - 5 \times 4 = 480 - 20 = 460$
$1368 = 460 \times 3 - 4 \times 3 = 1380 - 12 = 1368$
$2730 = 1368 \times 2 - 3 \times 2 = 2736 - 6 = 2730$
Applying the same pattern to the second series starting with $16$:
$(a) = 16 \times 6 - 7 \times 6 = 96 - 42 = 54$
$(b) = 54 \times 5 - 6 \times 5 = 270 - 30 = 240$
$(c) = 240 \times 4 - 5 \times 4 = 960 - 20 = 940$
$(d) = 940 \times 3 - 4 \times 3 = 2820 - 12 = 2808$
Therefore,the value in place of $(d)$ is $2808$.

(B) The pattern in the original series is:
$7 \times 13 = 91$
$91 \times 11 = 1001$
$1001 \times 7 = 7007$
$7007 \times 5 = 35035$
$35035 \times 3 = 105105$
The multipliers are prime numbers in descending order: $13, 11, 7, 5, 3$.
Applying the same pattern to the new series starting with $14.5$:
$(a) = 14.5 \times 13 = 188.5$
$(b) = 188.5 \times 11 = 2073.5$
$(c) = 2073.5 \times 7 = 14514.5$
Thus,the value at $(c)$ is $14514.5$.

(A) The pattern in the original series is as follows:
$582 - 2^3 = 582 - 8 = 574$
$574 + 3^3 = 574 + 27 = 601$
$601 - 4^3 = 601 - 64 = 537$
$537 + 5^3 = 537 + 125 = 662$
$662 - 6^3 = 662 - 216 = 446$
Applying the same pattern to the second series starting with $204$:
$(a) = 204 - 2^3 = 204 - 8 = 196$
$(b) = 196 + 3^3 = 196 + 27 = 223$
$(c) = 223 - 4^3 = 223 - 64 = 159$
$(d) = 159 + 5^3 = 159 + 125 = 284$
Thus,the value at $(d)$ is $284$.

(D) The pattern of the original series is as follows:
$85 \times 0.5 + 0.5 = 43$
$43 \times 1 + 1 = 44$
$44 \times 1.5 + 1.5 = 67.5$
$67.5 \times 2 + 2 = 137$
$137 \times 2.5 + 2.5 = 345$
Following the same logic for the second series starting with $125$:
$(a) = 125 \times 0.5 + 0.5 = 62.5 + 0.5 = 63$
$(b) = 63 \times 1 + 1 = 64$
$(c) = 64 \times 1.5 + 1.5 = 96 + 1.5 = 97.5$
Therefore,the value at $(c)$ is $97.5$.

(A) The pattern of the series is as follows:
$1 \times 2 + 2 \times 2 = 6$
$6 \times 4 + 4 \times 3 = 36$
$36 \times 6 + 6 \times 4 = 240$
$240 \times 8 + 8 \times 5 = 1960$
Following this logic,the next term is:
$1960 \times 10 + 10 \times 6 = 19600 + 60 = 19660$

(D) The given series follows a pattern of multiplying by consecutive decimals: $0.2, 0.3, 0.4, 0.5, 0.6$.
$949 \times 0.2 = 189.8$
$189.8 \times 0.3 = 56.94$
$56.94 \times 0.4 = 22.776$
$22.776 \times 0.5 = 11.388$
$11.388 \times 0.6 = 6.8328$
Thus,the missing number is $56.94$.

(D) The pattern follows the logic: $Term_{n+1} = Term_n \times (3 \times 2^{n-1}) + (1.5 \times 2^{n-1})$.
Step $1$: $14 \times 3 + 1.5 = 43.5$
Step $2$: $43.5 \times 6 + 3 = 264$
Step $3$: $264 \times 12 + 6 = 3174$
Step $4$: $3174 \times 24 + 12 = 76188$
Thus,the missing term is $3174$.

(C) The pattern follows the multiplication of the previous term by consecutive even numbers squared:
$41 \times 2^{2} = 41 \times 4 = 164$
$164 \times 4^{2} = 164 \times 16 = 2624$
$2624 \times 6^{2} = 2624 \times 36 = 94464$
$94464 \times 8^{2} = 94464 \times 64 = 6045696$
Thus,the missing number is $94464$.

(A) The pattern is based on multiplying the previous term by a factor that increases according to the sum of the previous two multipliers.
$12 \times 1 = 12$
$12 \times 1.5 = 18$
$18 \times (1 + 1.5) = 18 \times 2.5 = 45$
$45 \times (1.5 + 2.5) = 45 \times 4 = 180$
$180 \times (2.5 + 4) = 180 \times 6.5 = 1170$
$1170 \times (4 + 6.5) = 1170 \times 10.5 = 12285$
Thus,the next term is $12285$.

(A) The pattern of the given series is as follows:
$40280625 \div 55 = 732375$
$732375 \div 45 = 16275$
$16275 \div 35 = 465$
$465 \div 25 = 18.6$
$18.6 \div 15 = 1.24$
In each step,the divisor decreases by $10$ $(55, 45, 35, 25, 15, ...)$.
Therefore,the next divisor will be $15 - 10 = 5$.
Next term $= 1.24 \div 5 = 0.248$.