Question

Prove that, for every natural number n,(n+1)^2 - n^2=(n+1) + n, i.e the difference of squares of two consecutive natural numbers is equal to their sum. Give an example.​

Answer 1

Required Answer:-

Question:

• Prove that, for every natural n, (n + 1)² - n² = (n + 1) + n. Give an example.

Proof:

We have to prove that,

(n + 1)² - n² = (n + 1) + n

So, here the proof comes,

Taking LHS,

(n + 1)² - n²

= n² + 2 × n × 1 + 1 - n²

= n² + 2n + 1 - n²

= 2n + 1

= n + n + 1

= (n + 1) + n

Taking RHS,

= (n + 1) + n

So, LHS = RHS

Therefore, for every positive natural number, the difference of the squares of two consecutive numbers is equal to the sum of the two numbers. (Proved)

Example:

Suppose the two numbers are 11 and 10.

So,

11² - 10²

= 121 - 100

= 21

= 11 + 10

Hence, the statement is true for any positive natural number.

Let's take another example,

Suppose, the numbers are 20 and 21

So,

21² - 20²

= 441 - 400

= 41

= 20 + 21

Hence, the statement is true for these two numbers also (21 and 20).

Answer 2

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