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Question Number 136500 by rexford last updated on 22/Mar/21

Answered by Dwaipayan Shikari last updated on 22/Mar/21

$$a-2d+a-d+a+a+d+a+2d+a+3d=78$$$$\Rightarrow 6a+3d=78\Rightarrow 2a+d=26$$$$sincea-2d>1\Rightarrow a>2d+1Greatestpossiblewhena=2d+1$$$$2a+d=26\Rightarrow 5d+2=26\Rightarrow \Rightarrow d=\frac{24}{5}$$

Commented by rexford last updated on 22/Mar/21

$$whyisthata-2d>1?$$

Answered by MJS_new last updated on 23/Mar/21

$$a+a+d+a+2d+a+3d+a+4d+a+5d=6a+15d=78$$$$6a+15d=78$$$$2a+5d=26$$$$\Rightarrow d=2k$$$$2a+10k=26$$$$a+5k=13$$$$\left(1\right)k<0\Rightarrow a=13-5k\wedge -k\in {\mathbb{N}}^{★}\Rightarrow \mathrm{greatest}\mathrm{difference}=+\mathrm{\infty }$$$$\left(2\right)k=0\Rightarrow a=13\Rightarrow \mathrm{greatest}\mathrm{difference}=0$$$$\mathrm{we}\mathrm{reject}\left(1\right)\mathrm{and}\left(2\right)$$$$\left(3\right)k>0\Rightarrow k=\frac{13-a}{5}\Rightarrow a=3\wedge k=2\vee a=8\wedge k=1$$$$\iff$$$$a=3\wedge d=4\vee a=8\wedge d=2$$$$\Rightarrow \mathrm{maximal}\mathrm{difference}\mathrm{is}4$$

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