1 + 3 + 5 + 7 [Sum of first four odd numbers] = 16 = 4²
1 + 3 + 5 + 7 + 9 [Sum of first five odd numbers] = 25 = 5²
1 + 3 + 5 + 7 + 9 + 11 [Sum of first six odd numbers] = 36 = 6²

n = 5
∴ n + 1  = 6
∴ There are 2n = 2\(\times\)5 = 10 non perfect square numbers between 5² and 6²
These are 26, 27, 28, 29, 30, 31, 32, 33, 34 and 35.

When 'n' is a member of a Pythagorean
triplet, then the triplet is
n² - 1, 2n, n² + 1
or (13²- 1), (2 x 13), (13² + 1)
or (169 - 1), (26), (169 + 1)
or 168, 26 and 170

We have 169 = 13²
= Sum of first 13 odd numbers
= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25
= (1 + 25) + (3 + 23) + (5 + 21) + (7 + 19) + (9 + 17) + (11 + 15) + 13
= 26 + 26 + 26 + 26 + 26 + 26 + 13
= (6 x 26) + 13 = 156 + 13 = 169

We know that, between n2 and (n + 1)², there are 2n non-square numbers.
\(\therefore\) Between 27 and 28, there are 2 x 27, i.e, 54 numbers.

Here , n = 11
∴ \(\frac { { n }^{ 2 }-1 }{ 2 } =\frac { { 11 }^{ 2 }-1 }{ 2 } =\frac { 121-1 }{ 2 } \)
=\(\frac { 120 }{ 2 } \)= 60
and  \(\frac { { n }^{ 2 }+1 }{ 2 } =\frac { { 11 }^{ 2 }+1 }{ 2 } =\frac { 121+1 }{ 2 } \)
=\(\frac { 121+1 }{ 2 } \)=61
∴  11² = 60 + 61 = 121.

Sum of first 4 odd natural numbers is, 1+ 3 + 5 + 7 = 16.
Hence, 16 is a square of 4.

Square of \(0.4\times 0.4=\frac { 4 }{ 10 } \times \frac { 4 }{ 10 } =\frac { 16 }{ 100 } =0.16\)

Since, 8 \(\times\) 8=64. So, the units place of 168² will be 4.

\(\because \quad 6.1=\frac { 61 }{ 10 } \)
\(\therefore \quad { (6.1) }^{ 2 }={ \left( \frac { 61 }{ 10 } \right) }^{ 2 }=\frac { 61\times 61 }{ 10\times 10 } =\frac { 3721 }{ 100 } =37.21\)

4, 9,16,25,36 and 49 are six perfect squares, which lie between 1 and 50.

We know that, a number ends with 2, 3, 7 or 8 is neverapeIfea square.
The number 7928 ends with 8, which is not one of the end digits of 0, 1, 4, 5, 6 or 9, so it is not a perfect square.

We know that, a number ends with 2, 3, 7 or 8 is neverapeIfea square.
The number 23453 ends with 3, which is not one of the end digits of 0, 1, 4, 5, 6 or 9, so it is not a perfect square.

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