A-particular-A-P-has-a-positive-common-difference-and-is-such-that-for-any-three-adjacent-terms-three-times-the-sum-of-their-squares-exceed-the-square-of-their-sum-by-37-5-Find-the-common

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Question Number 167851 by otchereabdullai@gmail.com last updated on 27/Mar/22

$$\mathrm{A}\mathrm{particular}\mathrm{A}.\mathrm{P}\mathrm{has}\mathrm{a}\mathrm{positive}$$$$\mathrm{common}\mathrm{difference}\mathrm{and}\mathrm{is}\mathrm{such}\mathrm{that}$$$$\mathrm{for}\mathrm{any}\mathrm{three}\mathrm{adjacent}\mathrm{terms},\mathrm{three}$$$$\mathrm{times}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{their}\mathrm{squares}\mathrm{exceed}$$$$\mathrm{the}\mathrm{square}\mathrm{of}\mathrm{their}\mathrm{sum}\mathrm{by}37.5.\mathrm{Find}$$$$\mathrm{the}\mathrm{common}\mathrm{difference}.$$

Answered by mr W last updated on 27/Mar/22

$$saythetermsarea-d,a,a+d$$$$3[{(a-d)}^2+a^2+{(a+d)}^2]-{(a-d+a+a+d)}^2=37.5$$$$3(3a^2+2d^2)-{\left(3a\right)}^2=\frac{75}{2}$$$$d^2=\frac{75}{12}=\frac{25}{4}$$$$\Rightarrow d=\frac{5}{2}$$

Commented by otchereabdullai@gmail.com last updated on 27/Mar/22

$$\mathrm{Infact}\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{and}\mathrm{give}\mathrm{you}$$$$\mathrm{long}\mathrm{life}\mathrm{because}\mathrm{you}\mathrm{are}\mathrm{beneficial}$$$$\mathrm{to}\mathrm{the}\mathrm{world}!\mathrm{God}\mathrm{bless}\mathrm{you}!$$

Commented by peter frank last updated on 27/Mar/22

$$\mathrm{thanks}$$

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