Patterns in Numbers

$1+3+5+...+33=$              

Observe the pattern.

$$\begin{gathered} 7^2=49 \\ {67}^2=4489 \\ {667}^2=444889 \\ {6667}^2=44448889 \\ {66667}^2=4444488889 \\ {666667}^2=444444888889 \end{gathered}$$

Now, write the square of ${66666667}^2$, making use of the above pattern.

Observe the pattern.

$$\begin{gathered} 7^2=49 \\ {67}^2=4489 \\ {667}^2=444889 \\ {6667}^2=44448889 \\ {66667}^2=4444488889 \\ {666667}^2=444444888889 \end{gathered}$$

Now, write the square of ${6666667}^2$, making use of the above pattern.

Observe the pattern.

$$\begin{gathered} 1^2= 1 \\ {11}^2= 1 2 1 \\ {111}^2= 1 2 3 2 1 \\ {1111}^2= 1 2 3 4 3 2 1 \\ {11111}^2= 1 2 3 4 5 4 3 2 1 \end{gathered}$$

Now, write the square of ${1111111}^2$, making use of the above pattern.

Observe the pattern.

$$\begin{gathered} 1^2= 1 \\ {11}^2= 1 2 1 \\ {111}^2= 1 2 3 2 1 \\ {1111}^2= 1 2 3 4 3 2 1 \\ {11111}^2= 1 2 3 4 5 4 3 2 1 \end{gathered}$$

Now, write the square of ${111111}^2$, making use of the above pattern.

Do you think the sum of any two consecutive positive integers is perfect square of a number? Give examples to support your answer.

Express ${19}^2$ as the sum of two consecutive integers.

Express ${11}^2$ as the sum of two consecutive integers.

Express ${13}^2$ as the sum of two consecutive integers.

Express ${21}^2$ as the sum of two consecutive integers.

Write the next number of the given number sequence

$1, 4, 9, 16,$ _____

Without actual addition, find the sum of the following.

$1+3+5+7+9+11+13+15+17+19$

Without actual addition, find the sum of the following.

$1+3+5+7+9+11+13+15$

Without actual addition, find the sum of the following.

$1+3+5+7+9+11+13$

The square of an odd number can always be written as the sum of two consecutive natural numbers. Can the reverse statement be true? Is the sum of any two consecutive natural numbers a perfect square of a number?

Consider the square number $1089$, express it as the sum of two consecutive positive integers.

Consider the square number $289$, express it as the sum of two consecutive positive integers.

Consider the following square number $225$, express it as the sum of two consecutive positive integers.

Consider the following square number $441$, express it as the sum of two consecutive positive integers.

Observe the given table.

Write multiplications tables of $6$. In each table, circle the triangular numbers as shown in the example. Draw a box for square numbers. List them in columns as shown.