Question

b) Observe the pattern given below. Identify the constants and the variables, and form a
general rule for the number of squares in the pattern.​

Answer 1

answer. Consider calculating the square of 54. We know 54 = 50 + 4. So, 542 = (50 + 4)2 = (50 + 4) (50 + 4) = 50(50 + 4) + 4(50 + 4) = 502 + (50 × 4) + (4 × 50) + 42 = 2500 + 200 + 200 + 16 = 2916. This is similar to the identity (a + b)2 = a2 + b2 + 2ab.

Other Number Pattern in Squares

This pattern is associated with finding the square of the numbers having 5 in their unit’s place. For finding the square we have to multiply the number(s) except for those in unit’s place with the next coming number in hundreds and add 25.

Consider squaring 65. Now, 652 = 4225 = (6 × 7) hundreds + 25 = 4200 + 25 = 4225. Similarly, 1152 = 13225 = (11 × 12) hundreds + 25 = 13200 + 25 = 13225.

Pythagorean Triplets

There are some cases for which the sum of the squares of two numbers will give rise to the square of another number. The three numbers are the Pythagorean triplets. The examples of Pythagorean triplets are 3, 4 and 5; 5, 12 and 13; 6, 8 and 10 etc.

32 + 42 = 9 + 16 = 25 = 52.

52 + 122 = 25 + 144 = 169 = 132.

This gives rise to the Pythagoras Theorem in a right-angled triangle.

Number Pattern

But the real question is how can we find such triplets? Any natural number, n> 1 which satisfies the condition (2n) + (n2 – 1) 2 = (n2 + 1) 2 satisfies the condition for Pythagorean triplets. Here, 2n, n2 – 1, and n2 + 1 forms a Pythagorean triplet.