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Answer 1

$\Large{\bf{\green{\mathfrak{\dag{\underline{\underline{Question:-}}}}}}}$

1² + 2² + 2² = 3²

2² + 3³ + 6² = 7²

3² + 4² + 12² = 13²

4² + 5² + __² = 21²

5² + __² + 30² = 31²

6² + 7² + __² = __²

$\Large{\bf{\orange{\mathfrak{\dag{\underline{\underline{Answer:-}}}}}}}$

Here,

• In the 1st column we can observe that they are natural numbers from 1 to 6.

• In the next column we can observe that the numbers are starting from 2 and ending with 7.

• We can also observe that the common difference between the numbers is 1.

Therefore,

• 2nd column, 5th row answer is 6.

5² + 6² + 30² = 31²

Verifying:-

• 25 + 36 + 900 = 961

• 61 + 900 = 961

• 961 = 961

Hence, our answer is correct.

Now,

• Let's see 3rd column.

• In 3rd column, we can observe that the numbers are starting from 2 and ending with __ (empty).

• There is no common difference in the 3rd column.

But,

• If we relate 3rd column with 1st and 2nd column, we can observe that,

1st column x 2nd column = 3rd column.

• 1 x 2 = 2

• 2 x 3 = 6

• 3 x 4 = 12

Then,

• 4 x 5 = 20

In the same way,

• 5 x __ = 30

• __ = $\sf \dfrac{30}{5}$

• __ = 6

Continuing,

• 5 x 6 = 30

In the same way,

• 6 x 7 = 42

Therefore,

• 3rd column, 4th row answer is 20.

• 3rd column, 6th row answer is 42.

Veryfing:-

• 4² + 5² + 20² = 21²

• 16 + 25 + 400 = 441

• 41 + 400 = 441

• 441 = 441

Next,

• 6² + 7² + 42² = __

• 36 + 49 + 1764 = __

• 85 + 1764 = __

• 1849 = __

We don't know the value of 4th column, 6th row. Let's find it.

Now,

• On observing 4th coloum,

• It starts from 3 and ends with __ (empty).

• There is no common difference also.

But wait,

• On careful observation,

• 7 - 3 = 4

• 13 - 7 = 6

• 21 - 13 = 8

• 31 - 21 = 10

By, this we can understand that,

• Difference of these numbers results in a even number.

According to this,

• __ - 31 = 12

• __ = 12 + 31

• __ = 43

Therefore,

• 4th column, 6th row answer is 43.

There is also an another way,

• On careful observation of 3rd and 4th column we can say that,

3rd column + 1 = 4th column.

By this,

• 42 + 1 = 43

Anyway,

Verifying:-

• 6² + 7² + 42² = 1849

• 36 + 49 + 1764 = 1849

• 85 + 1764 = 1849

• 1849 = 1849

Coincidentally, our answer of 3rd column, 6th row i.e 42 has also verified.

$\Large{\bf{\red{\mathfrak{\dag{\underline{\underline{Final \: Answer:-}}}}}}}$

• 1² + 2² + 2² = 3²
• 2² + 3³ + 6² = 7²

• 3² + 4² + 12² = 13²

• 4² + 5² + 20² = 21²

• 5² + 6² + 30² = 31²

• 6² + 7² + 42² = 43²

Answer 2

Solution :-

By recognizing the pattern

For the fourth one

The last number are increasing by 2

Like

2

6

12

20

30

42

For the fifth one

The middle term are increasing by 1

Like

2

3

4

5

6

Hence in fourth the number is 6

For the answer of last

The number are increasing  by 2

Like

3

3 + 4 = 7

7 + 6 = 13

13 + 8 = 21

21 + 10 = 31

31 + 12 = 43