Patterns in Square Numbers

Express $196$ using the odd number sequence.

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 ${\left(10\right)}^2+{\left(11\right)}^2+{\left(12\right)}^2+{\left(13\right)}^2+.........+{\left(50\right)}^2$

$15$ is a triangular number.

Identify the triangular number among the given numbers: $5,11,21,25,30$

The seventh triangular number is the same as:

Prove that any repunit in base $9$ is a triangular number.

Without performing the addition, find the sum of $1+3+5+7+........+21$

Square of a natural number $n$ is equal to the sum of first _____ odd natural numbers

Without performing the addition, find the sum of $1+3+5+7+........+15$

Find the non-perfect square numbers between  ${\left(100\right)}^2$ and ${\left(101\right)}^2$

Express ${49}^2$ as the sum of two consecutive numbers.

[Hint: $9^2=81=40+41=\frac{9^2-1}{2}+\frac{9^2+1}{2}$]

Find the non-perfect square numbers between  ${\left(48\right)}^2$ and ${\left(49\right)}^2$

Given pattern:

$7^2=49=\left(\frac{7^2-1}{2}+\frac{7^2+1}{2}\right)=24+25$

Using the pattern express ${21}^2$ as the sum of two consecutive numbers.

Find the non-perfect square numbers between  ${\left(12\right)}^2$ and ${\left(13\right)}^2$

Find the missing number, if same rule is followed in all the three figures.

Write the next number of the given number sequence

$1, 4, 9, 16,$ _____

What is the tenth triangular number?

Find the sixteenth triangular number.

The ninth triangular number is

Find the sum of the fifth square number and the twelfth triangular number