16, 19, 21, 30, 46, 71, 107

Series Pattern
2 x 3 + 5 = 11, 11 x 4 - 6 = 38
38 x 5 + 7 = 197, 197 x 6 - 8 = 1174
Should come in place of 1172
1174 x 7 + 9 = 8827
8827 x 8 - 10 = 65806
Clearly, 1172 is wrong and will be replaced by 1174.

Series Pattern
(15 x 11), (15 x 13), (15 x 17), (15 x 19), (15 x 23), (15 x 29)
? Missing term = (15 x 23) = 345

The 1st term of an HP is 1/17 .
Hence, for the corresponding AP, the terms is 17 .
Here, a = 17
nth term of an AP is given by T_n = a + (n -1)d
? T₂ = 17 + d, T₄ = 17 + 3d
T₅ = 17 + 4d, T₆ = 17 + 5d
? (17 + d) (17 + 3d) = (17 + 4d) (17 + 5d)
? 289 + 51d + 17d + 3d² = 289 + 85d + 68d + 20d²
? 68d + 3d² = 153d + 20d²
? 17d² = -85d
? d = -5
? 3rd term is T₃ = a + 2d = 17 + 2(-5)
= 17 - 10 = 7
Hence, for the corresponding HP, the 3rd term is 1/7.

Let a be the first term and and r be the common ratio of the GP. From the given problem,
a + ar^{n - 1} = 66 ...(i)
Also, ar x ar^{n - 2} = 128 ? a²r^{n - 1} = 128 ... (ii)
From Eq. (ii),
a.ar^{n - 1} = 128
ar^{n - 1} = 128/ a
On substitution this in Eq.(i), we get
a + 128/a = 66
a² - 66a + 128 = 0
a = [-b ± ?b² - 4ac] /2a
= 66 ± ?66² - 4 x 128 x 1)/2 = 64 or 2

Put n = 1, now the sum of the series = 1.2.4 = 8
Put n = 1, in the options
(a) 1(1 + 1) (1 + 2) = 6
(b) 1(1 + 1)/12 x (3 + 19 + 26) = 8
(c) (1 + 1) (1 + 2) (1 + 3)/4 = 6
(d) 1(1 + 1) (1 + 2) (1 + 3 )/3 = 8
As, the sum of series = 8
Hence, option (a) and (c) can be rejected.
Now, put n = 2
Sum 1. 2. 4 + 2 . 3 . 5 = 38
Put n = 2, in option (b)
2(2 + 1)/12 (3 x 4 + 19 x 2 + 26)38
Hence, (b) is the correct option.

Series pattern 4 x 0.5 = 2
2 x 1.5 = 3
Should come in place of 3.5
3 x 2.5 = 7.5
7.5 x 3.5 = 26.25
26.25 x 4.5 = 118.125
Clearly, 3.5 is wrong and is replaced by 3.

Series pattern
20 + 1² = 21,
21 + 2² = 25,
25 + 3² = 34,
34 + 4² = 50,
50 + 5² = 75 Should come in place of ' ? '
75 + 6² = 111

Series pattern
62 + 5² = 62 + 25 = 87
87 + 10² = 87 + 100 = 187
187 + 15² = 187 + 225 = 412
412 + 20² = 412 + 400 = 812
812 + 25² = 812 + 625 = 1437
Should come in place of '?'

Series pattern
The elements of the given series are the numbers formed by joining together consecutive add numbers in order
i.e, 1 and 3, 3 and 5, 5 and 7, 7 and 9, 9 and 11,...
? Missing tern = Number formed by joining 11 and 13 = 1113

As we know that the angles are supplementary so sum of angles will be 180 degree.
Let us assume that the ratio factor is r.
According to question,
Angles are supplementary and have a ratio of 1:4.
r + 4r = 180
? 5r = 180
? r = 180/5
? r = 36