Series completion Questions in English

(D) The given series is $7, 14, 28, \ldots$.
Observing the pattern: $14/7 = 2$,$28/14 = 2$.
Since the ratio between consecutive terms is constant,this is a Geometric Progression $(G.P.)$.
Here,the first term $a = 7$ and the common ratio $r = 2$.
The formula for the $n^{th}$ term of a $G.P.$ is $T_n = a \times r^{(n-1)}$.
For the $10^{th}$ term $(n = 10)$:
$T_{10} = 7 \times 2^{(10-1)} = 7 \times 2^9$.
Since $2^9 = 512$,we have $T_{10} = 7 \times 512 = 3584$.

(B) The given series is $1, 4, 9, 16, 25, \ldots$
These numbers represent the squares of consecutive natural numbers:
$1^{2} = 1$
$2^{2} = 4$
$3^{2} = 9$
$4^{2} = 16$
$5^{2} = 25$
The next number in the series should be the square of $6$,which is $6^{2} = 36$.
Therefore,the missing number is $36$.

(C) The given series is $20, 19, 17, (\ldots), 10, 5$.
Let us observe the differences between consecutive terms:
$20 - 19 = 1$
$19 - 17 = 2$
Following this pattern of subtracting consecutive integers $(1, 2, 3, 4, 5)$,the next difference should be $3$.
Missing number $= 17 - 3 = 14$.
To verify,check the subsequent terms:
$14 - 4 = 10$
$10 - 5 = 5$
The pattern holds true. Therefore,the missing number is $14$.

(B) The given series is $2, 3, 5, 7, 11, (\dots), 17$.
Observing the sequence,these are consecutive prime numbers.
$A$ 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 $2, 3, 5, 7, 11, 13, 17, \dots$
Therefore,the missing number after $11$ is $13$.

(D) The given series is $6, 11, 21, 36, 56, \ldots$
Let us analyze the differences between consecutive terms:
$11 - 6 = 5$
$21 - 11 = 10$
$36 - 21 = 15$
$56 - 36 = 20$
The differences are in an arithmetic progression: $5, 10, 15, 20, \ldots$
The next difference should be $20 + 5 = 25$.
Therefore,the missing number is $56 + 25 = 81$.

(C) The given series is $1, 6, 13, 22, 33, \dots$
Let us find the difference between consecutive terms:
$6 - 1 = 5$
$13 - 6 = 7$
$22 - 13 = 9$
$33 - 22 = 11$
The differences follow a pattern of consecutive odd numbers starting from $5$: $5, 7, 9, 11, \dots$
The next difference should be $11 + 2 = 13$.
Therefore,the missing number is $33 + 13 = 46$.

(B) The given series is a geometric progression where each term is obtained by multiplying the preceding term by $3$.
$3 \times 3 = 9$
$9 \times 3 = 27$
$27 \times 3 = 81$
Following the same pattern,the next term is:
$81 \times 3 = 243$.

(A) The given series is $1, 9, 17, 33, 49, 73, ...$
Let's analyze the differences between consecutive terms:
$9 - 1 = 8$
$17 - 9 = 8$
$33 - 17 = 16$
$49 - 33 = 16$
$73 - 49 = 24$
The pattern of differences is $8, 8, 16, 16, 24, ...$
This indicates that each difference is repeated twice,and the difference increases by $8$ after every two steps.
Following this pattern,the next difference should be $24$.
Therefore,the missing number $= 73 + 24 = 97$.

(A) The given series is $2, 5, 9, (\ldots), 20, 27$.
Let us analyze the differences between consecutive terms:
$5 - 2 = 3$
$9 - 5 = 4$
The pattern of differences is increasing by $1$ each time $(+3, +4, +5, +6, \ldots)$.
Following this pattern,the next difference should be $+5$:
$9 + 5 = 14$.
To verify,the next difference should be $+6$:
$14 + 6 = 20$ (which matches the series).
Finally,$20 + 7 = 27$ (which also matches).
Therefore,the missing number is $14$.

(B) The pattern observed in the series is an addition of consecutive multiples of $4$:
$9 - 5 = 4$
$17 - 9 = 8$
$29 - 17 = 12$
$45 - 29 = 16$
Following this pattern,the next difference should be $16 + 4 = 20$.
Therefore,the missing number is $45 + 20 = 65$.

(C) Each number in the series is obtained by multiplying the preceding number by $2$ and then adding $1$.
Following this pattern:
$(3 \times 2) + 1 = 7$
$(7 \times 2) + 1 = 15$
$(15 \times 2) + 1 = 31$
$(31 \times 2) + 1 = 63$
Therefore,the missing number is:
$(63 \times 2) + 1 = 126 + 1 = 127$.

(D) Analyze the differences between consecutive terms:
$15 - 6 = 9$
$91 - 66 = 25$
$66 - 45 = 21$
The differences are $9, \ldots, \ldots, 21, 25$.
This indicates an arithmetic progression of differences increasing by $4$ $(9, 13, 17, 21, 25)$.
Therefore,the missing term is $15 + 13 = 28$ and $28 + 17 = 45$,which fits the pattern.
The correct alternative is $28$.

(C) Each term in the series is the sum of the two preceding terms.
Thus,$1 + 2 = 3$; $2 + 3 = 5$; $3 + 5 = 8$.
Following this pattern,the missing number is $5 + 8 = 13$.

(D) Each term of the series is obtained by multiplying the preceding term by $3$.
$0.5 \times 3 = 1.5$
$1.5 \times 3 = 4.5$
$4.5 \times 3 = 13.5$
$13.5 \times 3 = 40.5$
Therefore,the missing number is $40.5$.

(C) The given series is $121, 225, 361, \ldots$
These numbers are squares of consecutive odd numbers with a common difference of $4$ in their bases:
$11^{2} = 121$
$15^{2} = 225$
$19^{2} = 361$
The bases are $11, 15, 19, \ldots$ which form an arithmetic progression with a common difference of $4$.
The next base will be $19 + 4 = 23$.
Therefore,the missing number is $23^{2} = 529$.

(A) Let us analyze the pattern of the given series: $0, 2, 8, 14, (. . .), 34$.
The differences between consecutive terms are:
$2 - 0 = 2$
$8 - 2 = 6$
$14 - 8 = 6$
This does not show a simple arithmetic progression. Let us look at the terms again:
$0 = 1^2 - 1$
$2 = 2^2 - 2$
$8 = 3^2 - 1$
$14 = 4^2 - 2$
The pattern follows alternating subtractions of $1$ and $2$ from the squares of consecutive integers ($n^2 - 1$ for odd $n$,$n^2 - 2$ for even $n$).
Following this pattern,the next term $(n=5)$ should be $5^2 - 1 = 25 - 1 = 24$.
Checking the next term $(n=6)$: $6^2 - 2 = 36 - 2 = 34$,which matches the series.
Therefore,the missing number is $24$.

(D) The given sequence is a combination of two alternating series:
Series $I$: $19, 38, 114, \dots$
Series $II$: $2, 3, 4, \dots$
Analyzing Series $I$:
$19 \times 2 = 38$
$38 \times 3 = 114$
Following this pattern,the next term should be $114 \times 4 = 456$.
Therefore,the missing number is $456$.

(B) The given series is $1, 2, 3, 6, 9, 18, (...), 54$.
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 $1.5$ (which is $\frac{3}{2}$).
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 term is $27$.

(D) The pattern of the series is based on the addition of consecutive squares of natural numbers:
$5 - 4 = 1 = 1^2$
$9 - 5 = 4 = 2^2$
$18 - 9 = 9 = 3^2$
$34 - 18 = 16 = 4^2$
Following this pattern,the next difference should be $5^2 = 25$.
Therefore,the missing number is $34 + 25 = 59$.

(D) The pattern in the series is as follows:
$3 \times 2 = 6$
$6 \times 3 = 18$
$18 \times 4 = 72$
Following this pattern,the next operation should be multiplication by $5$.
Therefore,the missing number is $72 \times 5 = 360$.

(C) Each number in the series is the product of the digits of the preceding number.
Following this pattern:
$6 \times 6 = 36$
$3 \times 6 = 18$
$1 \times 8 = 8$
Therefore,the missing number is $8$.

(A) The pattern of the series is based on the addition of powers of $2$:
$25 - 21 = 4 = 2^2$
$33 - 25 = 8 = 2^3$
$49 - 33 = 16 = 2^4$
$81 - 49 = 32 = 2^5$
Following this pattern,the next difference should be $2^6 = 64$.
Therefore,the missing number is $81 + 64 = 145$.

(A) The given series is $12, 32, 72, 152, \ldots$
Let us analyze the differences between consecutive terms:
$32 - 12 = 20$
$72 - 32 = 40$
$152 - 72 = 80$
The differences are $20, 40, 80, \ldots$,which follow a pattern of doubling the previous difference $(20 \times 2 = 40, 40 \times 2 = 80)$.
The next difference should be $80 \times 2 = 160$.
Therefore,the missing number is $152 + 160 = 312$.

(D) The given series is a combination of two alternating series:
Series $I$: $3, 5, 7, 9$ (where each term increases by $2$)
Series $II$: $6, 20, 42, (...)$
Let's analyze the differences between consecutive terms in Series $II$:
$20 - 6 = 14$
$42 - 20 = 22$
The difference between the differences is $22 - 14 = 8$. Therefore,the next difference should be $22 + 8 = 30$.
Thus,the missing number is $42 + 30 = 72$.

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

(D) The series follows a pattern of increasing differences.
$15 - 2 = 13$
$41 - 15 = 26$
$80 - 41 = 39$
The differences are multiples of $13$: $13 \times 1, 13 \times 2, 13 \times 3, \ldots$
The next difference should be $13 \times 4 = 52$.
Therefore,the missing number is $80 + 52 = 132$.

(C) Let us analyze the differences between consecutive terms in the series:
$10 - 8 = 2$
$14 - 10 = 4$
$18 - 14 = 4$
Following the pattern of doubling the difference or repeating,let us check the latter part of the series:
$66 - 50 = 16$
$50 - 34 = 16$
This suggests the pattern of differences is $2, 4, 4, 8, 8, 16, 16$.
Applying this to the missing term:
$18 + 8 = 26$
Then,$26 + 8 = 34$,which matches the next term in the series.
Therefore,the missing number is $26$.

(C) The given series is $1, 2, 6, 24, \dots$
Observe the pattern between consecutive terms:
$1 \times 2 = 2$
$2 \times 3 = 6$
$6 \times 4 = 24$
The pattern follows multiplication by consecutive integers starting from $2$ (i.e.,$\times 2, \times 3, \times 4, \dots$).
Following this pattern,the next term should be multiplied by $5$:
$24 \times 5 = 120$.
Therefore,the missing number is $120$.

(D) Each term in the series is obtained by squaring the preceding term and subtracting $1$.
Step $1$: $(2)^2 - 1 = 4 - 1 = 3$
Step $2$: $(3)^2 - 1 = 9 - 1 = 8$
Step $3$: $(8)^2 - 1 = 64 - 1 = 63$
Step $4$: $(63)^2 - 1 = 3969 - 1 = 3968$
Therefore,the missing term is $3968$.

(B) Analyze the difference between consecutive terms in the series:
$115.5 - 95 = 20.5$
$138 - 115.5 = 22.5$
The differences are increasing by $2$ at each step $(20.5, 22.5, 24.5, 26.5, \dots)$.
Therefore,the next difference should be $22.5 + 2 = 24.5$.
Adding this to the last known term: $138 + 24.5 = 162.5$.
To verify,the next term should be $162.5 + 26.5 = 189$,which matches the series.
Thus,the missing term is $162.5$.

(B) The pattern of the series is as follows:
$4 \times 3 - 2 = 10$
$10 \times 3 - 2 = 28$
$28 \times 3 - 2 = 82$
$82 \times 3 - 2 = 244$
$244 \times 3 - 2 = 730$
Therefore,the missing number is $28$.

(D) The given series is $4, 32, 128, (\ldots)$.
Observe the relationship between consecutive terms:
$4 \times 8 = 32$
$32 \times 4 = 128$
The multipliers are $8$ and $4$,which are decreasing by a factor of $2$ $(8/2 = 4)$.
Following this pattern,the next multiplier should be $4/2 = 2$.
Therefore,the missing term is $128 \times 2 = 256$.

(B) The pattern followed in the series is alternating between $\times 2 + 1$ and $\times 2 - 1$.
Step $1$: $2 \times 2 + 1 = 5$
Step $2$: $5 \times 2 - 1 = 9$
Step $3$: $9 \times 2 + 1 = 19$
Step $4$: $19 \times 2 - 1 = 37$
Step $5$: Following the pattern,the next operation is $\times 2 + 1$.
Missing number $= 37 \times 2 + 1 = 74 + 1 = 75$.

(B) The given series is $24, 60, 120, 210, \dots$
Let us find the differences between consecutive terms:
$60 - 24 = 36$
$120 - 60 = 60$
$210 - 120 = 90$
The differences are $36, 60, 90, \dots$
Now,let us find the differences between these differences:
$60 - 36 = 24$
$90 - 60 = 30$
The second-order differences are $24, 30, \dots$,which increase by $6$.
The next second-order difference should be $30 + 6 = 36$.
Therefore,the next first-order difference should be $90 + 36 = 126$.
Thus,the missing number is $210 + 126 = 336$.

(C) The given series is $165, 195, 255, 285, 345, (...)$.
Each term can be expressed as $15$ multiplied by a prime number:
$165 = 15 \times 11$
$195 = 15 \times 13$
$255 = 15 \times 17$
$285 = 15 \times 19$
$345 = 15 \times 23$
The sequence of prime numbers used is $11, 13, 17, 19, 23$. The next prime number after $23$ is $29$.
Therefore,the missing term is $15 \times 29 = 435$.

(D) The given series is $5, 17, 37, 65, (\ldots), 145$.
We can observe the pattern as follows:
$5 = 2^{2} + 1$
$17 = 4^{2} + 1$
$37 = 6^{2} + 1$
$65 = 8^{2} + 1$
The pattern follows the form $n^{2} + 1$ where $n$ is an even number starting from $2$.
Following this pattern,the next term should be $10^{2} + 1$.
$10^{2} + 1 = 100 + 1 = 101$.
The term after that is $12^{2} + 1 = 144 + 1 = 145$,which matches the last term of the series.
Therefore,the missing number is $101$.

(B) The given series follows the pattern where each term is the sum of the two preceding terms (Fibonacci-like sequence).
$9 + 11 = 20$
$11 + 20 = 31$
$20 + 31 = 51$
$31 + 51 = 82$
Therefore,the missing number is $51$.

(D) The differences between consecutive terms are:
$16 - 5 = 11$
$49 - 16 = 33$
$104 - 49 = 55$
The pattern of differences is $11, 33, 55, \dots$
These are multiples of $11$: $(11 \times 1), (11 \times 3), (11 \times 5), \dots$
The next difference should be $(11 \times 7) = 77$.
Therefore,the missing number is $104 + 77 = 181$.

(D) The pattern of the series is as follows:
$34 \div 2 + 1 = 17 + 1 = 18$
$18 \div 2 + 1 = 9 + 1 = 10$
$10 \div 2 + 1 = 5 + 1 = 6$
$6 \div 2 + 1 = 3 + 1 = 4$
$4 \div 2 + 1 = 2 + 1 = 3$
Therefore,the missing term is $3$.

(C) The given series is $462, 420, 380, (...), 306$.
Let us observe the differences between consecutive terms:
$462 - 420 = 42$
$420 - 380 = 40$
The pattern of differences is decreasing by $2$ each time (i.e.,$-42, -40, -38, -36, \dots$).
Following this pattern,the next difference should be $-38$:
$380 - 38 = 342$
To verify,the next difference should be $-36$:
$342 - 36 = 306$
Since this matches the last term,the missing number is $342$.

(A) The pattern follows the rule: multiply the previous term by $3$ and subtract an increasing integer starting from $1$.
Step $1$: $3 \times 3 - 1 = 8$
Step $2$: $8 \times 3 - 2 = 22$
Step $3$: $22 \times 3 - 3 = 63$
Step $4$: $63 \times 3 - 4 = 185$
Step $5$: $185 \times 3 - 5 = 550$
Therefore,the missing number is $550$.

(C) The pattern followed in the series is as follows:
$1 \times 2 + 0 = 2$
$2 \times 2 + 1 = 5$
$5 \times 2 + 2 = 12$
$12 \times 2 + 3 = 27$
$27 \times 2 + 4 = 58$
$58 \times 2 + 5 = 121$
Following this logic,the next term is:
$121 \times 2 + 6 = 242 + 6 = 248$.

(C) The pattern of the series is as follows:
$0.5 + 0.05 = 0.55$
$0.55 + 0.10 = 0.65$
$0.65 + 0.15 = 0.80$
Following this pattern,the next increment should be $0.20$.
Therefore,the missing number is $0.8 + 0.20 = 1$.

(A) The pattern followed in the series is:
$1st$ term: $3$
$2nd$ term: $8$
$3rd$ term: $13$
$4th$ term: $24$
$5th$ term: $41$
Let's analyze the relationship between the terms:
$3 + 8 + 2 = 13$ ($3rd$ term)
$8 + 13 + 3 = 24$ ($4th$ term)
$13 + 24 + 4 = 41$ ($5th$ term)
The pattern is: $(\text{Term}_{n} + \text{Term}_{n+1} + n) = \text{Term}_{n+2}$,where $n$ is the index of the first term in the addition.
To find the $6th$ term:
$24 + 41 + 5 = 70$
Therefore,the missing term is $70$.

(A) Analyze the differences between consecutive terms:
$97 - 86 = 11$
$86 - 73 = 13$
$73 - 58 = 15$
$58 - 45 = 13$
The pattern of differences is $-11, -13, -15, -13, \ldots$
Following this alternating pattern,the next difference should be $-11$.
Therefore,the missing number $= 45 - 11 = 34$.

(A) The given series consists of consecutive prime numbers starting from $17$.
The prime numbers are numbers greater than $1$ that have only two factors: $1$ and the number itself.
The sequence is: $17, 19, 23, 29, \dots, 37$.
The next prime number after $29$ is $31$.
Therefore,the missing number is $31$.

(B) The pattern of the series is based on the addition of consecutive integers.
The differences between consecutive terms are:
$6 - 5 = 1$
$9 - 6 = 3$
$15 - 9 = 6$
The differences are $1, 3, 6, \ldots$,which follow the pattern of triangular numbers or the sum of consecutive integers: $1, (1+2), (1+2+3), \ldots$
Following this logic,the next difference should be $(1+2+3+4) = 10$.
Therefore,the missing number is $15 + 10 = 25$.
To verify,the next difference should be $(1+2+3+4+5) = 15$,and $25 + 15 = 40$,which matches the final term in the series.

(A) The given series is $3, 12, 27, 48, 75, 108, \dots$
We can observe that each term follows the pattern $3 \times n^{2}$,where $n$ is the position of the term:
$3 \times 1^{2} = 3 \times 1 = 3$
$3 \times 2^{2} = 3 \times 4 = 12$
$3 \times 3^{2} = 3 \times 9 = 27$
$3 \times 4^{2} = 3 \times 16 = 48$
$3 \times 5^{2} = 3 \times 25 = 75$
$3 \times 6^{2} = 3 \times 36 = 108$
The next term in the series corresponds to $n = 7$:
Missing number $= 3 \times 7^{2} = 3 \times 49 = 147$.

(B) Each term is obtained by adding $111$ to the preceding term.
$134 + 111 = 245$
$245 + 111 = 356$
$356 + 111 = 467$
Therefore,the missing number is:
$467 + 111 = 578$

(D) The pattern follows the rule: $\text{Term}_{n+1} = \text{Term}_n \times 2 + n$,where $n$ is the incrementing value starting from $1$.
Step $1$: $6 \times 2 + 1 = 13$
Step $2$: $13 \times 2 + 2 = 28$
Step $3$: $28 \times 2 + 3 = 59$
Therefore,the missing number is $59$.