Solve the following questions: 1. n2 + (n+1)2 + (n(n+1))2 = (n(n+1)+1)2. Fill in missing terms. 2. Sum of first n odd numbers. 3. Express a square number as sum of consecutive odd numbers.

The given pattern is $n^2 + (n+1)^2 + (n(n+1))^2 = (n(n+1)+1)^2$.

• For $4^2 + 5^2 + \_ = 21^2$: Here $n=4$. The missing term is $(4 \times 5)^2 = 20^2$. The right side is $(4 \times 5 + 1)^2 = 21^2$. So, $4^2 + 5^2 + \mathbf{20^2} = 21^2$.

• For $5^2 + \_ + 30^2 = 31^2$: Here $n=5$. The second term is $(n+1)^2 = (5+1)^2 = 6^2$. The third term is $(5 \times 6)^2 = 30^2$. The right side is $(5 \times 6 + 1)^2 = 31^2$. So, $5^2 + \mathbf{6^2} + 30^2 = 31^2$.

• For $6^2 + 7^2 + \_ = \_ ^2$: Here $n=6$. The third term is $(6 \times 7)^2 = 42^2$. The right side is $(6 \times 7 + 1)^2 = 43^2$. So, $6^2 + 7^2 + \mathbf{42^2} = \mathbf{43^2}$.

The sum of the first $n$ odd numbers is $n^2$.

• (i) $1+3+5+7+9$ There are 5 odd numbers.

  $$1+3+5+7+9 = 5^2 = \boxed{25}$$

• (ii) $1+3+5+7+9+11+13+15+17+19$ There are 10 odd numbers.

  $$1+3+5+7+9+11+13+15+17+19 = 10^2 = \boxed{100}$$

• (iii) $1+3+5+7+9+11+13+15+17+19+21+23$ There are 12 odd numbers.

  $$1+3+5+7+9+11+13+15+17+19+21+23 = 12^2 = \boxed{144}$$

The sum of the first $n$ odd numbers is $n^2$.

• (i) Express 49 as the sum of 7 odd numbers. Since $49 = 7^2$, it is the sum of the first 7 odd numbers.

  $$49 = \boxed{1+3+5+7+9+11+13}$$

• (ii) Express 121 as the sum of 11 odd numbers. Since $121 = 11^2$, it is the sum of the first 11 odd numbers.

  $$121 = \boxed{1+3+5+7+9+11+13+15+17+19+21}$$

The number of non-square numbers between $n^2$ and $(n+1)^2$ is $2n$.

• (i) Between 12 and 13: Here $n=12$. Number of numbers $= 2 \times 12 = \boxed{24}$

• (ii) Between 25 and 26: Here $n=25$. Number of numbers $= 2 \times 25 = \boxed{50}$

• (iii) Between 99 and 100: Here $n=99$. Number of numbers $= 2 \times 99 = \boxed{198}$