The term of the series are (1² + 1)(2² + 1)(3² + 1)(4² + 1)(5² + 1)(6² + 1)(7² + 1),..............
So, 64 is wrong and must be replaced by (8² + 1) i.e., 65

Each term of the series is the product of the preceding two terms.
So, 5000 is wrong and must be replaced by (50 × 500) = i.e., 25000

We have :
I: 5       27       61       122       213       340     509
II:  22       34       61         91       127     169
III:        12        27        30        36       42
IV:            15        3         6   &nbsp ;       6           
Clearly, the given series becomes a triangular pattern series if each term in IV: is 6.
Again, in II: 34 must be replaced by (61 - 24) i.e., 37 and 22 by (37 - 18) i.e., 19.
Thus, in the given series, 27 is wrong and must be replaced by (5 + 19) i.e., 24.

The terms of the series are (2³ - 3), (3³ - 3), (4³ - 3), (5³ - 3), (6³ - 3), (7³ -3), (8³ -3).
So, 27 is wrong and must be replaced (3³ - 3) i.e., 24

The given sequence is ¹⁰⁰/₉ , ²⁵/₂ , ¹⁰⁰/₇ , ⁵⁰/₃, ............i.e., ¹⁰⁰/₉, ¹⁰⁰/₈, ¹⁰⁰/₇, ¹⁰⁰/₆.................
So, missing term = ¹⁰⁰/₅ = 20

The correct pattern is × 2 + 1.
So, 39 is wrong and must be replaced by (15 × 2 + 1) i.e., 31.

The correct sequence is 3, 3² , 3³,4, 4² , 4³,5, 5² , 5³
So. 10 is wrong and must replaced by 3² i.e., 9

The given sequence is a combination of three series:
I. 0, 3, 6
II. 4, 7, ?
III. 6, 9, 12
The pattern in each of these series is + 3
So, missing term = 7 + 3 = 10

The given sequence is a combination of two series:
I. 3, 7, 13, 21, 31, ? and
II. 4, 7, 13, 22, 34
The pattern in I is + 4, + 6, + 8, + 10,......
The pattern in II is + 3, + 6, + 9, + 12,..............
So, missing term = 31 + 12 = 43

Clearly, 90 = 30 × 3, 180 = 6 × 30, 12 = 2 × 6, 50 = 25 × 2, 100 = 4 × 25, 200 = 50 × 4
So, missing term = 3 × 50 = 150

The terms of the series are product of two consecutive prime numbers i.e., (2 × 3), (3 × 5), (5 × 7), (7 × 11),...........
So, 165 is wrong and must be replaced by (11 × 13) i.e., 143.

Each term in the series is obtained by adding 22 to the preceding term.
So, 186 is wrong and must be replaced by (165 + 22) i.e., 187.

The given sequence is a combination of three series:
I. 1st, 4th, 7th, 10th, terms i.e., 8, 7, 6, ?
II. 2nd, 5th, 8th, 11th terms i.e.,9, 10, 11, 12
III. 3rd, 6th, 9th, terms i.e., 8, 9, 10
The pattern in I. is - 1 So, the missing term is = 6 - 1 = 5

Clearly, 5 + 3 = 8, 8 + 3 = 11, 11 + 3 = 14,................
So, the series in an A.P. in which a = 5 and d = 3
Let 320 be the nth term of the series.
Then, 320 = 5 + (n - 1) × 3 or (n - 1) = 105 or n = 106

The given series consists of squares of consecutive odd numbers i.e., 1² , 3² , 5² , 7²,..........
So. missing term = 9² = 81

The given series is a triangular pattern series.
So, we have
2       12       36       80     150     ?
     10       24       44       70
         14        20      26
              6           6
So, missing term
= 150 + (70 + 26 + 6)
= 150 + 102
= 252

The given sequence is a combination of two series:
I. 625, 125, 25, 5 and
II. 5, 25, ?
The pattern is ÷ 5, while that in II is × 5.
So, missing term = 25 × 5 = 125

Clearly, 3 + 6 = 9, 9 + 6 = 15,.............
So, the series is an A.P in which a = 3 and d = 6
∴21st term = a + (21 - 1)d = a + 20d
= 3 +20 × 6 = 123.

Clearly, the numerators of the given fractions are consecutive natural numbers. So, the numerator of the missing fraction should be 6. Also, the denominator of each fraction is multiplied by √5 to obtain the denominator of the next fraction. So. the denominator of the missing fraction should be 25√5. Hence,the missing term is ⁶/_25√5

Clearly, 2 × 3 = 6, 6 × 3 = 18, 18 × 3 = 54,..............
S, the series is a G.P. in which a = 2, r = 3
∴8th term = ar^{(8 - 1)}= ar⁷ = 2 × 3⁷ = (2 × 2187) = 4374

The given series consists of numbers each of which, on dividing by 7, leaves a remainder 3, No other number except 346 satisfies the property.

The given sequence is a combination of two series:
I. 13, 24, 35, 46, 57 and
II. 32, 43, ?, 65, 76
The pattern in both I and II is + 11
So, missing term = 43 + 11 = 54

The sequence in the numerators is + 5, + 10, + 20, ......and that in the denominators is + 11, + 22, + 44,............
So, the numerator of the missing fraction should be (9 + 10)i.e., 19 and the denominator should be (20 + 22)i.e., 42. Thus, the missing term is ¹⁹/₄₂ .

Each term of the series is obtained by dividing the preceding term by 4.
So, 1536 is wrong and must be replaced by (1536 ÷ 4) i.e., 384.

The correct pattern is + 6, + 8, + 10, + 12, + 14.
So, 45 is wrong and must be replaced by (30 + 10) i.e., 40.

The pattern is × ³/₂.
So, missing number = 13¹/₂ × ³/₂ = ²⁷/₂ × ³/₂ = ⁸¹/₄ = 20¹/₄.

The given series is a triangular pattern series.
So, we have
Series I :  2     9     28     ?
                126   217    344
Series II :  7     19     ?
                ?   91    127
Series III :       12     ?     ?
                ?   36
Clearly, to form a pattern, the missing terms in Series III must be 18, 24, 30
So, missing term
= 127 + (61 + 24 + 6)
= 127 + 91
= 218

The pattern is - 1, × 10 + 1, - 1, × 10 + 1, - 1, × 10 +1, ....
So, the missing term = 10 × 10 + 1 = 101

Clearly, 5 × 2 = 10, 10 × 2 =20, 20 × 2 = 40,.............................
So, the series is G.P. in which a = 5 and r = 2.
Let 1280 be the nth term of the series.
Then, 5 × 2^{n - 1}= 1280 ⇔2^{n - 1}= 256 = 2⁸ ⇔ n - 1 = 8 ⇔ n = 9

The given sequence is a combination of two series:
I. 0, 3, 8, 15, 24, ? and
II. 2, 5, 10, 17, 26
The pattern in each one of I and II is + 3, + 5, + 7, + 9,..............
So, missing term = 24 + 11 = 35