Question

Fill the given boxes​

Answer 1

STEP-BY-STEP EXPLANATION:

Given that the pattern and the number of line segments of equal length used

(i) 5n + 1

(ii) 2n + 1

(iii) 4n + 1

(iv) 5n + 1

while 'n' denotes the number of shapes formed.

We need to find out the number of segments required for making (i) 6 such digits, 9 such digits (ii) 50 equilateral triangle, 100 equilateral triangle (iii) 16 such letters, 35 such letters (iv) 5 rectangles, 20 rectangles.

Now,

(i) For 6 such digits (n = 6)

→ 5n + 1

→ 5(6) + 1

→ 30 + 1

→ 31

For 9 such digits (n = 9)

→ 5(9) + 1

→ 45 + 1

→ 46

(ii) For 50 equilateral triangles (n = 50)

→ 2n + 1

→ 2(50) + 1

→ 100 + 1

→ 101

For 100 equilateral triangles (n = 100)

→ 2(100) + 1

→ 200 + 1

→ 201

(iii) For 16 such letters (n = 16)

→ 4n + 1

→ 4(16) + 1

→ 64 + 1

→ 65

For 35 such letters

→ 4(35) + 1

→ 140 + 1

→ 141

(iv) For 5 rectangles (n = 5)

→ 5n + 1

→ 5(5) + 1

→ 25 + 1

→ 26

For 20 rectangles (n = 20)

→ 5(20) + 1

→ 100 + 1

→ 101

Answer 2

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✯ Hint ✯

Note that all the integers given to us are odd integers and 30 is an even integer. Use the fact that the sum of three odd numbers cannot be an even number, or, the sum of three odd numbers is always an odd number, to arrive at the final answer.

☆ Complete step-by-step answer ☆

In this question, we are given the equation _+_+_=30

We need to fill the boxes in this equation using only (1, 3, 5, 7, 9, 11, 13, 15) to make it correct.

As we can see from the numbers available to us: (1, 3, 5, 7, 9, 11, 13, 15). All of these numbers are odd numbers.

We need to use these numbers to form 30 which is an even number.

We already know that the sum of three odd numbers cannot be an even number.

Or, the sum of three odd numbers is always an odd number.

So, using the integers available to us (all odd integers), we cannot satisfy the given equation and form 30 (an even integer).

Hence, it is not possible to directly add three of the available numbers to satisfy the given equation.

✯ Note ✯

Though, it is not possible to directly add three of the available numbers to satisfy the given equation, we can use a trick to get an answer which satisfies the given equation. We can use the factorial function of one of the available numbers (which will be even) to fill the boxes and satisfy the equation. One of such examples is the following: _+_+_=30. This satisfies the given equation and we have used only the integers available to us.