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If-a-1-a-2-a-3-be-an-AP-then-prove-that-n-1-2m-1-n-1-a-n-2-m-2m-1-a-n-2-a-2m-2-




Question Number 108553 by I want to learn more last updated on 17/Aug/20
If    a_1 ,  a_2 ,  a_3 ,    be an AP, then prove that:     Σ_(n  =  1) ^(2m)  (− 1)^(n  −  1)  a_n ^2     =   (m/(2m  −  1))(a_n ^2    −  a_(2m) ^2 )
$$\mathrm{If}\:\:\:\:\mathrm{a}_{\mathrm{1}} ,\:\:\mathrm{a}_{\mathrm{2}} ,\:\:\mathrm{a}_{\mathrm{3}} ,\:\:\:\:\mathrm{be}\:\mathrm{an}\:\mathrm{AP},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\:\:\:\underset{\mathrm{n}\:\:=\:\:\mathrm{1}} {\overset{\mathrm{2m}} {\sum}}\:\left(−\:\mathrm{1}\right)^{\mathrm{n}\:\:−\:\:\mathrm{1}} \:\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \:\:\:\:=\:\:\:\frac{\mathrm{m}}{\mathrm{2m}\:\:−\:\:\mathrm{1}}\left(\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \:\:\:−\:\:\mathrm{a}_{\mathrm{2m}} ^{\mathrm{2}} \right) \\ $$

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