Patterns in Numbers

Given that ;

$1^2+2^2=\frac{2\times 3\times 5}{6}, 1^2+2^2+3^2=\frac{3\times 4\times 7}{6}, 1^2+2^2+3^2+4^2=\frac{4\times 5\times 9}{6} \mathrm{etc}$

Using the above pattern, determine ${\left(10\right)}^2+{\left(11\right)}^2+{\left(12\right)}^2+{\left(13\right)}^2+.........+{\left(50\right)}^2$

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.

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.

Find the sum without actually adding the following odd numbers: $1+3+5+7+\dots \dots +35$

Find the sum without actually adding the first $99$ odd natural numbers.

Write ${11}^2$ as the sum of consecutive odd natural numbers.

Write ${10}^2$ as the sum of consecutive odd natural numbers.

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