Question 7

We can find the count of whole numbers between two consecutive perfect squares using a simple pattern.

Step 1 — Discovering the rule

We want to count the whole numbers that lie strictly between the square of a natural number $n$ and the square of the next natural number, $(n+1)$. Let us test with small numbers first to see a pattern. Consider $n=2$. The two consecutive natural numbers are 2 and 3. Their squares are $2^2 = 4$ and $3^2 = 9$. The whole numbers strictly between 4 and 9 are 5, 6, 7, 8. There are 4 such numbers. Notice that $4 = 2 \times 2$. This is $2n$.

Consider $n=3$. The two consecutive natural numbers are 3 and 4. Their squares are $3^2 = 9$ and $4^2 = 16$. The whole numbers strictly between 9 and 16 are 10, 11, 12, 13, 14, 15. There are 6 such numbers. Notice that $6 = 2 \times 3$. This is $2n$.

This pattern suggests that for any natural number $n$, the count of whole numbers between $n^2$ and $(n+1)^2$ is $2n$. Let us prove this general rule using algebra. The whole numbers we are counting start from $n^2 + 1$. They end at $(n+1)^2 - 1$. To find the total count of integers from $A$ to $B$ (inclusive), we use the formula $B - A + 1$. Here, $A = n^2 + 1$ and $B = (n+1)^2 - 1$. Number of whole numbers $= ((n+1)^2 - 1) - (n^2 + 1) + 1$ $= (n^2 + 2n + 1 - 1) - n^2 - 1 + 1$ $= n^2 + 2n - n^2$

$\boxed{\text{Number of whole numbers} = 2n}$ This rule tells us that there are $2n$ whole numbers between $n^2$ and $(n+1)^2$. These numbers are not perfect squares because $n^2$ and $(n+1)^2$ are consecutive perfect squares.

Step 2 — Solving for 16 and 17

We need to find the numbers between the squares of 16 and 17. Here, our first natural number $n$ is 16. The next natural number is $(n+1) = 17$. Using our rule, the count of whole numbers is $2n$. Number of values $= 2 \times 16$

$\boxed{32}$

Step 3 — Solving for 99 and 100

We need to find the numbers between the squares of 99 and 100. Here, our first natural number $n$ is 99. The next natural number is $(n+1) = 100$. Using our rule, the count of whole numbers is $2n$. Number of values $= 2 \times 99$

$\boxed{198}$

Answer

(i) 32 (ii) 198