Series completion Questions in English

(D) The given sequence follows a pattern of alternating operations: $+2$ and $\times 2$.
Step $1$: $4 + 2 = 6$
Step $2$: $6 \times 2 = 12$
Step $3$: $12 + 2 = 14$
Step $4$: $14 \times 2 = 28$
Step $5$: $28 + 2 = 30$
Following this pattern,the next operation is $\times 2$:
Step $6$: $30 \times 2 = 60$
Therefore,the missing number is $60$.

(B) The given series is a geometric progression where each term is obtained by multiplying the previous term by a common ratio of $-2$.
Step $1$: $4 \times (-2) = -8$
Step $2$: $-8 \times (-2) = 16$
Step $3$: $16 \times (-2) = -32$
Step $4$: $-32 \times (-2) = 64$
Step $5$: $64 \times (-2) = -128$
Therefore,the missing term is $-128$.

(B) The given series is $2, 7, 14, 23, ?, 47$.
Calculate the differences between consecutive terms:
$7 - 2 = 5$
$14 - 7 = 7$
$23 - 14 = 9$
The differences follow an arithmetic progression of odd numbers: $5, 7, 9, \dots$
The next difference should be $9 + 2 = 11$.
Therefore,the missing number is $23 + 11 = 34$.
To verify,the next term would be $34 + 13 = 47$,which matches the series.

(C) The given series is $2, 6, 11, 17, (\ldots), 32$.
Let us observe the differences between consecutive terms:
$6 - 2 = 4$
$11 - 6 = 5$
$17 - 11 = 6$
The pattern of differences is an increasing sequence of consecutive integers: $+4, +5, +6, \ldots$
Following this pattern,the next difference should be $+7$.
Therefore,the missing term is $17 + 7 = 24$.
To verify,the next term should be $24 + 8 = 32$,which matches the last term of the series.

(C) Analyze the differences between consecutive terms in the series:
$10 - 3 = 7$
$20 - 10 = 10$
$33 - 20 = 13$
$49 - 33 = 16$
$68 - 49 = 19$
The differences are $7, 10, 13, 16, 19$,which form an arithmetic progression with a common difference of $3$.
Following this pattern,the next difference should be $19 + 3 = 22$.
Therefore,the missing number is $68 + 22 = 90$.

(B) The pattern in the series is that each number is obtained by multiplying the preceding number by $2$ and adding $1$.
Step $1$: $3 \times 2 + 1 = 7$
Step $2$: $7 \times 2 + 1 = 15$
Step $3$: $15 \times 2 + 1 = 31$
Step $4$: $31 \times 2 + 1 = 63$
Step $5$: $63 \times 2 + 1 = 126 + 1 = 127$
Therefore,the next number in the series is $127$.

(C) The given sequence is a combination of two alternating series:
Series $1$: $6, 12, 18, 24$ (multiples of $6$)
Series $2$: $24, (\ldots), 8$ (multiples of $8$ in descending order: $24, 16, 8$)
Thus,the missing number is $16$.

(D) The given series is $212, 179, 146, 113, (\ldots)$.
Observe the difference between consecutive terms:
$212 - 179 = 33$
$179 - 146 = 33$
$146 - 113 = 33$
Since the difference between consecutive terms is constant at $33$,we subtract $33$ from the last term to find the missing term.
$\therefore$ Missing term $= 113 - 33 = 80$.

(C) The series is arranged in groups of three consecutive terms where the middle term is the product of the two end terms.
Group $1$: $2, 6, 3 \implies 2 \times 3 = 6$
Group $2$: $4, 20, 5 \implies 4 \times 5 = 20$
Group $3$: $6, (\ldots), 7 \implies 6 \times 7 = 42$
Therefore,the missing number is $42$.

(B) Let us analyze the pattern of the series: $2, 5, 9, 19, 37, \ldots$
Step $1$: $2 \times 2 + 1 = 5$
Step $2$: $5 \times 2 - 1 = 9$
Step $3$: $9 \times 2 + 1 = 19$
Step $4$: $19 \times 2 - 1 = 37$
The pattern follows an alternating sequence of multiplying by $2$ and adding $1$,followed by multiplying by $2$ and subtracting $1$.
Step $5$: Since the previous operation was subtracting $1$,the next operation must be adding $1$.
$37 \times 2 + 1 = 74 + 1 = 75$.
Therefore,the correct alternative is $75$.

(D) The given series is $2, 6, 12, 20, 30, 42, 56, (\ldots)$.
We can observe the pattern as follows:
$1 \times 2 = 2$
$2 \times 3 = 6$
$3 \times 4 = 12$
$4 \times 5 = 20$
$5 \times 6 = 30$
$6 \times 7 = 42$
$7 \times 8 = 56$
The pattern follows the product of consecutive integers $n \times (n+1)$.
Therefore,the next term is $8 \times 9 = 72$.

(A) This series is obtained by alternatively multiplying by $3$ and dividing by $2$.
Following the pattern:
$8 \times 3 = 24$
$24 \div 2 = 12$
$12 \times 3 = 36$
$36 \div 2 = 18$
$18 \times 3 = 54$
Therefore,the next step is to divide by $2$:
$54 \div 2 = 27$.
Thus,the missing term is $27$.

(C) The given series is: $165, 195, 255, 285, 345, \ldots$
We can express these terms as multiples of $15$ with consecutive prime numbers:
$165 = 15 \times 11$
$195 = 15 \times 13$
$255 = 15 \times 17$
$285 = 15 \times 19$
$345 = 15 \times 23$
The prime numbers used are $11, 13, 17, 19, 23$. The next prime number after $23$ is $29$.
Therefore,the missing term is $15 \times 29 = 435$.

(A) The given series follows an alternating pattern of adding $5$ and subtracting $7$.
Step-by-step analysis:
$71 + 5 = 76$
$76 - 7 = 69$
$69 + 5 = 74$
$74 - 7 = 67$
$67 + 5 = 72$
Following this pattern,the next operation is to subtract $7$ from the last term:
$72 - 7 = 65$
Therefore,the missing term is $65$.

(B) The series follows an alternating pattern of adding $3$ and subtracting $1$.
Step-by-step analysis:
$9 + 3 = 12$
$12 - 1 = 11$
$11 + 3 = 14$
$14 - 1 = 13$
Following this pattern,the next step is to add $3$ to the last term:
$13 + 3 = 16$
Checking the next term: $16 - 1 = 15$,which matches the series.
Therefore,the missing term is $16$.

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

(C) Analyze the differences between consecutive terms:
$10 - 3 = 7$
$20 - 10 = 10$
$33 - 20 = 13$
$49 - 33 = 16$
$68 - 49 = 19$
The differences follow an arithmetic progression with a common difference of $3$: $7, 10, 13, 16, 19, \ldots$
The next difference in the series will be $19 + 3 = 22$.
Therefore,the missing number is $68 + 22 = 90$.

(C) The sum of any three consecutive terms of the series gives the next term.
$1 + 3 + 4 = 8$
$3 + 4 + 8 = 15$
$4 + 8 + 15 = 27$
Following this pattern,the missing number is the sum of the three terms preceding it:
$8 + 15 + 27 = 50$
Therefore,the correct option is $C$.

(D) Each number in the series is obtained by calculating the product of the digits of the preceding number.
Step $1$: The first number is $66$. The product of its digits is $6 \times 6 = 36$.
Step $2$: The second number is $36$. The product of its digits is $3 \times 6 = 18$.
Step $3$: The third number is $18$. The product of its digits is $1 \times 8 = 8$.
Therefore,the missing number is $8$.

(C) The given series is $11, 13, 17, 19, 23, 25, \ldots$
Analyzing the differences between consecutive terms:
$13 - 11 = 2$
$17 - 13 = 4$
$19 - 17 = 2$
$23 - 19 = 4$
$25 - 23 = 2$
The pattern of differences is $+2, +4, +2, +4, +2, \ldots$
Following this pattern,the next difference should be $+4$.
Therefore,the missing number is $25 + 4 = 29$.

(C) The difference between consecutive numbers increases by $1$ in each step.
The pattern is as follows:
$2 + 2 = 4$
$4 + 3 = 7$
$7 + 4 = 11$
$11 + 5 = 16$
Following this pattern,the next difference should be $6$.
$\therefore$ The missing number $= 16 + 6 = 22$.

(C) The given series is $0, 2, 6, (\ldots), 20, 30, 42$.
Let us observe the differences between consecutive terms:
$2 - 0 = 2$
$6 - 2 = 4$
The pattern of differences is increasing by $2$ $(2, 4, 6, 8, 10, 12, \ldots)$.
Following this pattern,the next difference after $4$ should be $6$.
So,the missing term is $6 + 6 = 12$.
To verify,the next term would be $12 + 8 = 20$,then $20 + 10 = 30$,and $30 + 12 = 42$,which matches the series.
Therefore,the correct alternative is $12$.

(A) The pattern in the series is that the numbers are alternately multiplied by $2$ and increased by $3$.
Step-by-step analysis:
$5 \times 2 = 10$ (Wait,the series is $5, 16, 13, 26, 29, 58, 61, \dots$)
Let's re-evaluate the pattern:
$5 + 11 = 16$
$16 - 3 = 13$
$13 + 13 = 26$
$26 + 3 = 29$
$29 + 29 = 58$
$58 + 3 = 61$
Actually,the pattern is: add $11$,subtract $3$,add $13$,add $3$,add $29$,add $3$. This is inconsistent.
Let's look at the pattern again: $5 \times 2 + 6 = 16$,$16 - 3 = 13$,$13 \times 2 = 26$,$26 + 3 = 29$,$29 \times 2 = 58$,$58 + 3 = 61$.
Following the pattern of multiplying by $2$ and adding $3$ alternately starting from $13$:
$13 \times 2 = 26$
$26 + 3 = 29$
$29 \times 2 = 58$
$58 + 3 = 61$
$61 \times 2 = 122$.
Therefore,the missing number is $122$.

(C) The given sequence is $2, 9, 28, 65, 126, \ldots$
Observing the pattern,each term follows the rule $n^{3} + 1$,where $n$ is the position of the term:
$1^{3} + 1 = 1 + 1 = 2$
$2^{3} + 1 = 8 + 1 = 9$
$3^{3} + 1 = 27 + 1 = 28$
$4^{3} + 1 = 64 + 1 = 65$
$5^{3} + 1 = 125 + 1 = 126$
Following this pattern,the next term is $6^{3} + 1 = 216 + 1 = 217$.

(A) The given series follows a pattern where each term is the sum of the two preceding terms.
$4 + 9 = 13$
$9 + 13 = 22$
$13 + 22 = 35$
Following this pattern,the next term is the sum of the last two terms:
$22 + 35 = 57$
Therefore,the missing number is $57$.

(B) The given series is $1, 8, 27, 64, 125, 216, (\ldots)$.
Observing the terms,we can see that they are cubes of consecutive natural numbers:
$1^{3} = 1$
$2^{3} = 8$
$3^{3} = 27$
$4^{3} = 64$
$5^{3} = 125$
$6^{3} = 216$
Following this pattern,the next term should be $7^{3}$.
$7^{3} = 7 \times 7 \times 7 = 343$.
Therefore,the missing number is $343$.

(D) The given series is $1, 2, 3, 6, 9, 18, (\ldots), 54$.
Observe the pattern between consecutive terms:
$1 \times 2 = 2$
$2 \times 1.5 = 3$
$3 \times 2 = 6$
$6 \times 1.5 = 9$
$9 \times 2 = 18$
The pattern follows alternating multiplication by $2$ and $1.5$ (or $\frac{3}{2}$).
Following this pattern,the next operation after multiplying by $2$ is to multiply by $1.5$:
$18 \times 1.5 = 27$.
To verify,the next term should be $27 \times 2 = 54$,which matches the series.
Therefore,the missing number is $27$.

(A) The given series is: $11, 13, 17, 19, 23, 29, 31, 37, 41, (\ldots)$
Observing the numbers,we can see that each term in the series is a consecutive prime number.
$A$ prime number is a natural number greater than $1$ that has no positive divisors other than $1$ and itself.
The prime number immediately following $41$ is $43$.
Therefore,the missing number is $43$.

(C) The given series is $2, 5, 11, 23, 47, \ldots$
Observe the pattern between consecutive terms:
$5 - 2 = 3$
$11 - 5 = 6$
$23 - 11 = 12$
$47 - 23 = 24$
The differences are $3, 6, 12, 24, \ldots$,which form a geometric progression where each term is multiplied by $2$.
The next difference should be $24 \times 2 = 48$.
Therefore,the missing number is $47 + 48 = 95$.

(B) The given sequence is a combination of two alternating series:
Series $1$: $4, 5, 7, 8, 10$ (where the pattern is $+1, +2, +1, +2, \ldots$)
Series $2$: $9, 12, 15, (\ldots)$ (where the pattern is $+3$)
Following the pattern of Series $2$,the next term is $15 + 3 = 18$.
Therefore,the missing term is $18$.

(D) The given sequence consists of two alternating series:
Series $1$: $10, 13, 16, 19$ (each term increases by $3$)
Series $2$: $5, 10, 20, (\ldots)$ (each term is multiplied by $2$)
Therefore,the missing number in the second series is $20 \times 2 = 40$.

(D) The given series is $2, 6, 12, 20, 30, 42, 56, \ldots$
We can observe the pattern as follows:
$1 \times 2 = 2$
$2 \times 3 = 6$
$3 \times 4 = 12$
$4 \times 5 = 20$
$5 \times 6 = 30$
$6 \times 7 = 42$
$7 \times 8 = 56$
The pattern follows the form $n \times (n+1)$ for $n = 1, 2, 3, \ldots$
Therefore,the next term is $8 \times 9 = 72$.

(B) The given series consists of prime numbers starting from $2$.
$A$ prime number is a natural number greater than $1$ that has no positive divisors other than $1$ and itself.
The sequence of prime numbers is $2, 3, 5, 7, 11, 13, 17, \ldots$
Therefore,the missing number after $11$ is $13$.

(C) The given series is $3, 8, 15, 24, 34, 48, 63$.
Let us calculate the differences between consecutive terms:
$8 - 3 = 5$
$15 - 8 = 7$
$24 - 15 = 9$
$34 - 24 = 10$
$48 - 34 = 14$
$63 - 48 = 15$
The pattern of differences should be consecutive odd numbers starting from $5$ (i.e.,$5, 7, 9, 11, 13, 15$).
If we replace $34$ with $35$,the differences become:
$35 - 24 = 11$
$48 - 35 = 13$
$63 - 48 = 15$
Thus,the sequence follows the pattern of adding consecutive odd numbers. Therefore,$34$ is the wrong number in the series.

(D) The pattern of the series is that each term is obtained by multiplying the preceding term by $3$ and then subtracting $4$ from the result.
Let us verify the terms:
$10 \times 3 - 4 = 30 - 4 = 26$
$26 \times 3 - 4 = 78 - 4 = 74$
$74 \times 3 - 4 = 222 - 4 = 218$
$218 \times 3 - 4 = 654 - 4 = 650$
$650 \times 3 - 4 = 1950 - 4 = 1946$
$1946 \times 3 - 4 = 5838 - 4 = 5834$
Comparing this with the given series,the term $654$ is incorrect,as it should be $650$.

(D) The pattern in the series is that each term is obtained by multiplying the preceding term by $2$ and subtracting $2$.
Let's check the terms:
$8 \times 2 - 2 = 16 - 2 = 14$
$14 \times 2 - 2 = 28 - 2 = 26$
$26 \times 2 - 2 = 52 - 2 = 50$
$50 \times 2 - 2 = 100 - 2 = 98$
$98 \times 2 - 2 = 196 - 2 = 194$
$194 \times 2 - 2 = 388 - 2 = 386$
Comparing this with the given series $8, 14, 26, 48, 98, 194, 386$,we see that $48$ is the wrong term,as it should be $50$.

(D) Let us analyze the differences between consecutive terms:
$13 - 8 = 5$
$21 - 13 = 8$
$32 - 21 = 11$
$47 - 32 = 15$
$63 - 47 = 16$
$83 - 63 = 20$
The pattern of differences is $5, 8, 11, 14, 17, 20$.
Looking at the differences: $5, 8, 11, 15, 16, 20$.
If we follow the pattern of adding $3$ to the difference $(5, 8, 11, 14, 17, 20)$:
$8 + 5 = 13$
$13 + 8 = 21$
$21 + 11 = 32$
$32 + 14 = 46$ (Instead of $47$)
$46 + 17 = 63$
$63 + 20 = 83$
Thus,$47$ is the wrong term.

(A) The pattern of the series is based on the formula $x_{n+1} = (x_n \times 2) + 1$.
Let us verify the terms:
$3 \times 2 + 1 = 7$
$7 \times 2 + 1 = 15$
$15 \times 2 + 1 = 31$ (Instead of $39$)
$31 \times 2 + 1 = 63$
$63 \times 2 + 1 = 127$
$127 \times 2 + 1 = 255$
$255 \times 2 + 1 = 511$
Since $39$ does not fit the pattern,it is the wrong term. The correct term should be $31$.

(C) The pattern followed in the series is that each term is obtained by subtracting $3$ from the preceding term and then dividing the result by $2$.
Let us check the terms:
$221 = (445 - 3) / 2 = 442 / 2 = 221$
$109 = (221 - 3) / 2 = 218 / 2 = 109$
$53 = (109 - 3) / 2 = 106 / 2 = 53$
In the given series,the term $46$ is incorrect because the expected value is $53$.
Continuing with $53$:
$25 = (53 - 3) / 2 = 50 / 2 = 25$
$11 = (25 - 3) / 2 = 22 / 2 = 11$
$4 = (11 - 3) / 2 = 8 / 2 = 4$
Therefore,$46$ is the wrong term.

(D) The pattern of the series is based on the addition of consecutive squares:
$1 + 1^2 = 2$
$2 + 2^2 = 6$
$6 + 3^2 = 15$
$15 + 4^2 = 31$
$31 + 5^2 = 56$
$56 + 6^2 = 92$
Comparing this with the given series,the term $91$ is incorrect.
The correct term should be $92$.

(D) The given series follows the pattern $n^{2} + 1$,where $n$ is the position of the term starting from $n = 1$.
$1^{2} + 1 = 2$
$2^{2} + 1 = 5$
$3^{2} + 1 = 10$
$4^{2} + 1 = 17$
$5^{2} + 1 = 26$
$6^{2} + 1 = 37$
$7^{2} + 1 = 50$
$8^{2} + 1 = 65$
Comparing this with the given series,the term $64$ is incorrect. The correct term should be $65$.

(C) Let us observe the pattern of division in the series:
$46080 \div 12 = 3840$
$3840 \div 10 = 384$
$384 \div 8 = 48$
$48 \div 6 = 8$
$8 \div 4 = 2$
$2 \div 2 = 1$
Comparing this with the given series,the term $24$ is incorrect.
The correct term should be $8$.

(C) Let us analyze the differences between consecutive terms:
$52 - 51 = 1$
$51 - 48 = 3$
$48 - 43 = 5$
$43 - 34 = 9$
$34 - 27 = 7$
$27 - 16 = 11$
The pattern of differences should be consecutive odd numbers: $1, 3, 5, 7, 9, 11$.
Following this pattern:
$52 - 1 = 51$
$51 - 3 = 48$
$48 - 5 = 43$
$43 - 7 = 36$ (Instead of $34$)
$36 - 9 = 27$
$27 - 11 = 16$
Thus,the term $34$ is incorrect and should be $36$.

(D) Let us analyze the differences between consecutive terms in the series:
$325 - 259 = 66$
$259 - 202 = 57$
$202 - 160 = 42$
$160 - 127 = 33$
$127 - 105 = 22$
$105 - 94 = 11$
The pattern of differences should be multiples of $11$ in descending order: $66, 55, 44, 33, 22, 11$.
If we apply this pattern starting from $325$:
$325 - 66 = 259$
$259 - 55 = 204$ (Instead of $202$)
$204 - 44 = 160$
$160 - 33 = 127$
$127 - 22 = 105$
$105 - 11 = 94$
Thus,the wrong term is $202$.

(D) The pattern of the series is as follows:
$125 + 1 = 126$
$126 - 2 = 124$
$124 + 3 = 127$
$127 - 4 = 123$
$123 + 5 = 128$
Comparing this with the given series $(125, 126, 124, 127, 123, 129)$,we see that the last term $129$ is incorrect.
The correct term should be $128$.

(D) The pattern of the series is as follows:
$3 \times 1 + 1 = 4$
$4 \times 2 + 2 = 10$
$10 \times 3 + 3 = 33$
$33 \times 4 + 4 = 136$
$136 \times 5 + 5 = 685$
$685 \times 6 + 6 = 4116$
Comparing this with the given series $3, 4, 10, 32, 136, 685, 4116$,we see that $32$ is the wrong term,as it should be $33$.

(A) The given series is $25, 36, 49, 81, 121, 169, 225$.
These numbers can be written as squares of integers:
$5^{2} = 25$
$6^{2} = 36$
$7^{2} = 49$
$9^{2} = 81$
$11^{2} = 121$
$13^{2} = 169$
$15^{2} = 225$
Observing the pattern,the sequence consists of squares of odd numbers starting from $5$,except for $36$,which is the square of an even number $(6^{2})$. Therefore,$36$ is the wrong term in the series.

(D) Analyze the difference between consecutive terms in the series:
$72 - 56 = 16$
$90 - 72 = 18$
$110 - 90 = 20$
$132 - 110 = 22$
$150 - 132 = 18$
The pattern of differences is an arithmetic progression: $16, 18, 20, 22, 24, \dots$
Following this pattern,the difference after $132$ should be $24$.
Therefore,the correct term should be $132 + 24 = 156$.
Since $150$ is present in the series instead of $156$,$150$ is the wrong term.

(D) The given series is $6, 13, 18, 25, 30, 37, 40$.
Let us observe the pattern of differences between consecutive terms:
$13 - 6 = 7$
$18 - 13 = 5$
$25 - 18 = 7$
$30 - 25 = 5$
$37 - 30 = 7$
$40 - 37 = 3$
The pattern of differences is $+7, +5, +7, +5, +7, \dots$
Following this pattern, the next difference after $7$ should be $5$.
Therefore, the term after $37$ should be $37 + 5 = 42$.
Since the last term is $40$ instead of $42$, $40$ is the wrong term.

(D) The pattern of the series is as follows:
$1st$ term: $10$
$2nd$ term: $10 + 4 = 14$
$3rd$ term: $14 \times 2 = 28$
$4th$ term: $28 + 4 = 32$
$5th$ term: $32 \times 2 = 64$
$6th$ term: $64 + 4 = 68$
$7th$ term: $68 \times 2 = 136$
In the given series,the last term is $132$,which is incorrect. The correct term should be $136$.