Puzzle : Sequence of numbers 

From the given equations,

(a+b+c)2=a2+b2+c2+2(ab+bc+ca)
(6)2=14 + 2(ab + bc + ca)
36 - 14 = 2(ab + bc + ca)
22/2 = ab + bc + ca
ab + bc + ca = 11

a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
= (6) (14 - 11)
=(6)(3)
=18

Since, a3 + b3 + c3 - 3abc = 18

36 - 3abc = 18

-3abc = 18 - 36

-3abc = -18

abc = 6.

Here the answer is 98.

The logic in the given puzzle:

2+3 = 8
3+7 = (7+2)x3 = 27
4+5 = (5+3)x4 = 32
5+8 = (8+4)x5 = 60
6+7 = (7+5)x6 = 72

Similarly,

7+8 = (8+6)x7 = 98.

The answer here is 1.

Logic Followed in the given puzzle ::

The logic here is the number of circles in each 4 digit number mentioned.

8801 has 5

5261 has 1

and so on

The Answer is 46.

Logic ::

First number on the right hand side is the second number in the left side of equation ( and ) second number on the right side is the sum of two numbers on the left side of the equation.

Hence, 2 + 4 = 4(2+4) = 46.

A + B = 15 .... (1)

B x C = 84 .... (2)

C ÷ 4 = 3  .... (3)

B + D = 18 .... (4)

From (3), we get C = 12

From (2), we get B = 84/12 = 7

From (1), we get A = 15 - 7 = 8

From (4), we get D = 18 - 7 = 11

Hence, D = 11.

The logic in this puzzle is summing the result of [subtraction of upper two digits] & [addition of lower two digits] 

Hence, the result is  

(5 - 3) + (2 + 8) = 2 + 10 = 12.

A - A - A = 20
B - B - B = 15
C - C - C = 25

A = -20
B = -15
C = -25

=> A + B + C = (-20)+(-15)+(-25) = -20 - 15 - 25 = -60.

5 + 5 = 10

5x 2 + 2 = 12

5 x 2 - 1 x 5 = 5

Hence, ∆ = 1.