Let ,
P(n) be the given equation,
Let us check for
![\[n\text{ }=\text{ }1,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5862da84ea43160bab4f0f449a446eec_l3.png "Rendered by QuickLaTeX.com")
![\[P\text{ }\left( 1 \right):\text{ }{{\left( 2.1\text{ }\text{ }1 \right)}^{2}}~=\text{ }1/3\text{ }\times \text{ }1\text{ }\times \text{ }\left( 4\text{ }\text{ }1 \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-295cdf8182978984d06fe88ead805a6a_l3.png "Rendered by QuickLaTeX.com")
![\[:\text{ }1\text{ }=\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c3368fc4179516c8efb169708a520795_l3.png "Rendered by QuickLaTeX.com")
P (n) is true for
![\[n\text{ }=\text{ }1.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d668617d54d95931aa3a863afbdeb96a_l3.png "Rendered by QuickLaTeX.com")
Now, let us check for P (n) is true for
![\[n\text{ }=\text{ }k\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bfb789f2d270d6be635d5f44f30de643_l3.png "Rendered by QuickLaTeX.com")
, and have to prove that P (k + 1) is true.
![\[P\text{ }\left( k \right):\text{ }{{1}^{2}}~+\text{ }{{3}^{2}}~+\text{ }{{5}^{2}}~+\text{ }\ldots \text{ }+\text{ }{{\left( 2k\text{ }\text{ }1 \right)}^{2}}~=\text{ }1/3\text{ }k\text{ }\left( 4{{k}^{2}}~\text{ }1 \right)\text{ }\ldots \text{ }\left( i \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-dfa7bb3d7656cbb845130ec26fc03fb6_l3.png "Rendered by QuickLaTeX.com")
So,
![\[{{1}^{2}}~+\text{ }{{3}^{2}}~+\text{ }{{5}^{2}}~+\text{ }\ldots \text{ }+\text{ }{{\left( 2k\text{ }\text{ }1 \right)}^{2}}~+\text{ }{{\left( 2k\text{ }+\text{ }1 \right)}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-27254991d98f8a669caa8995f09a3a96_l3.png "Rendered by QuickLaTeX.com")
Now, substituting the value of P (k) we get,
![\[=\text{ }1/3\text{ }k\text{ }\left( 4{{k}^{2}}~\text{ }1 \right)\text{ }+\text{ }{{\left( 2k\text{ }+\text{ }1 \right)}^{2}}~\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e379de662424a9bf0c3a79a1b5c83217_l3.png "Rendered by QuickLaTeX.com")
by using equation (i)
![\[=\text{ }1/3\text{ }k\text{ }\left( 2k\text{ }+\text{ }1 \right)\text{ }\left( 2k\text{ }\text{ }1 \right)\text{ }+\text{ }{{\left( 2k\text{ }+\text{ }1 \right)}^{2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4448b15e8d8520c72e74f6805ecfac50_l3.png "Rendered by QuickLaTeX.com")
or,
![\[=\text{ }\left( 2k\text{ }+\text{ }1 \right)\text{ }\left[ \left\{ k\left( 2k-1 \right)/3 \right\}\text{ }+\text{ }\left( 2k+1 \right) \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1dfda2d055b64691a6426cffec292cea_l3.png "Rendered by QuickLaTeX.com")
or,
![\[=\text{ }\left( 2k\text{ }+\text{ }1 \right)\text{ }\left[ 2{{k}^{2}}~\text{ }k\text{ }+\text{ }3\left( 2k+1 \right) \right]\text{ }/\text{ }3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bff8f4c704526d571a3d81221923017f_l3.png "Rendered by QuickLaTeX.com")
or,
![\[=\text{ }\left( 2k\text{ }+\text{ }1 \right)\text{ }\left[ 2{{k}^{2}}~\text{ }k\text{ }+\text{ }6k\text{ }+\text{ }3 \right]\text{ }/\text{ }3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7747cea2e970ceca93e600c69e92c52c_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }\left[ \left( 2k+1 \right)\text{ }2{{k}^{2}}~+\text{ }5k\text{ }+\text{ }3 \right]\text{ }/3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3554a4fb3bb890e9096580f0593bbb41_l3.png "Rendered by QuickLaTeX.com")
or,
![\[=\text{ }\left[ \left( 2k+1 \right)\text{ }\left( 2k\left( k+1 \right) \right)\text{ }+\text{ }3\text{ }\left( k+1 \right) \right]\text{ }/3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bfc216617a0c590fdc9639afd14df4c2_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }\left[ \left( 2k+1 \right)\text{ }\left( 2k+3 \right)\text{ }\left( k+1 \right) \right]\text{ }/3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-689838af09072fa5eadcf2e472dfe662_l3.png "Rendered by QuickLaTeX.com")
or,
![\[=\text{ }\left( k+1 \right)/3\text{ }\left[ 4{{k}^{2}}~+\text{ }6k\text{ }+\text{ }2k\text{ }+\text{ }3 \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-aa9d55c6a7ce0dca36c9ba8c1203550f_l3.png "Rendered by QuickLaTeX.com")
![\[=\text{ }\left( k+1 \right)/3\text{ }\left[ 4{{k}^{2}}~+\text{ }8k\text{ }\text{ }1 \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-36e2962d11b85de570bd000212175e0c_l3.png "Rendered by QuickLaTeX.com")
or,
![\[=\text{ }\left( k+1 \right)/3\text{ }\left[ 4{{\left( k+1 \right)}^{2}}~\text{ }1 \right]\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-95bbc17375a4070e0c1480f986b75d85_l3.png "Rendered by QuickLaTeX.com")
P (n) is true for
![\[n\text{ }=\text{ }k\text{ }+\text{ }1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6384bad4271818becb835911febc8d76_l3.png "Rendered by QuickLaTeX.com")
Hence, P (n) is true for all n ∈ N