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given-the-sum-of-the-first-n-terms-of-an-A-P-is-n-2-the-sum-of-of-the-first-2n-terms-of-the-same-A-P-is-n-2-n-show-that-the-sum-of-the-first-4n-terms-is-4n-2-8n-4-

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Question Number 34707 by mondodotto@gmail.com last updated on 10/May/18
given the sum of the first n terms of an A.P is n^2 the sum of of the first 2n terms of the same A.P is n^2 +n. show that the sum of the first 4n terms is 4n^2 −8n+4.
$$\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{A}}.\boldsymbol{\mathrm{P}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}}\:\mathrm{2}\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\: \\ $$$$\boldsymbol{\mathrm{same}}\:\boldsymbol{\mathrm{A}}.\boldsymbol{\mathrm{P}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{n}}^{\mathrm{2}} +\boldsymbol{\mathrm{n}}. \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{first}} \\ $$$$\mathrm{4}\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{is}}\:\mathrm{4}\boldsymbol{\mathrm{n}}^{\mathrm{2}} −\mathrm{8}\boldsymbol{\mathrm{n}}+\mathrm{4}. \\ $$
Commented by Rasheed.Sindhi last updated on 10/May/18
If sum of n terms of an AP=n^2 then the sum of 2n terms of the same AP=(2n)^2 =4n^2 ≠n^2 +n The sum of 4n terms=(4n)^2 =16n^2 The sum of 4n terms ≠ 4n^2 −8n+4
$$\:\mathrm{If}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{n}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{AP}=\mathrm{n}^{\mathrm{2}} \\ $$$$\:\mathrm{then} \\ $$$$\:\:\:\:\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{2n}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{AP}=\left(\mathrm{2n}\right)^{\mathrm{2}} =\mathrm{4n}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\neq\mathrm{n}^{\mathrm{2}} +\mathrm{n} \\ $$$$\:\:\:\:\:\:\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{4n}\:\mathrm{terms}=\left(\mathrm{4n}\right)^{\mathrm{2}} =\mathrm{16n}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{4n}\:\mathrm{terms}\:\:\neq\:\mathrm{4}\boldsymbol{\mathrm{n}}^{\mathrm{2}} −\mathrm{8}\boldsymbol{\mathrm{n}}+\mathrm{4} \\ $$
Commented by mondodotto@gmail.com last updated on 10/May/18
please recheck
$$\mathrm{please}\:\mathrm{recheck} \\ $$

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