The-fifth-nineth-sixteenth-terms-of-a-linear-sequence-are-consecutive-terms-of-an-exponential-sequence-1-Find-the-common-difference-of-the-linear-sequence-in-terms-of-the-first-term-2-Show-t

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Question Number 10489 by Saham last updated on 13/Feb/17

$$\mathrm{The}\mathrm{fifth},\mathrm{nineth},\mathrm{sixteenth}\mathrm{terms}\mathrm{of}\mathrm{a}\mathrm{linear}$$$$\mathrm{sequence}\mathrm{are}\mathrm{consecutive}\mathrm{terms}\mathrm{of}\mathrm{an}\mathrm{exponential}$$$$\mathrm{sequence}.$$$$\left(1\right)\mathrm{Find}\mathrm{the}\mathrm{common}\mathrm{difference}\mathrm{of}\mathrm{the}\mathrm{linear}$$$$\mathrm{sequence}\mathrm{in}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{term}$$$$\left(2\right)\mathrm{Show}\mathrm{that}{21}^{\mathrm{th}},{37}^{\mathrm{th}},{65}^{\mathrm{th}}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{linear}$$$$\mathrm{sequence}\mathrm{are}\mathrm{consecutive}\mathrm{terms}\mathrm{of}\mathrm{an}\mathrm{exponential}$$$$\mathrm{sequence}\mathrm{whose}\mathrm{common}\mathrm{ratio}\mathrm{is}\frac{3}{4}$$

Answered by mrW1 last updated on 14/Feb/17

$$\left(1\right)$$$$a_5=a+4d$$$$a_9=a+8d$$$$a_{16}=a+15d$$$$\frac{a+15d}{a+8d}=\frac{a+8d}{a+4d}$$$$letx=\frac{d}{a}$$$$\frac{1+15x}{1+8x}=\frac{1+8x}{1+4x}$$$$1+19x+60x^2=1+16x+64x^2$$$$4x^2-3x=0$$$$4x(x-\frac{3}{4})=0$$$$x=0or$$$$x=\frac{3}{4}$$$$x=\frac{d}{a}=\frac{3}{4}$$$$d=\frac{3}{4}a$$$$\left(2\right)$$$$a_{21}=a+20d=a+20\times \frac{3}{4}a=16a$$$$a_{37}=a+36d=a+36\times \frac{3}{4}a=28a$$$$a_{65}=a+64d=a+64\times \frac{3}{4}a=49a$$$$\frac{a_{65}}{a_{37}}=\frac{49a}{28a}=\frac{7}{4}$$$$\frac{a_{37}}{a_{21}}=\frac{28a}{16a}=\frac{7}{4}=\frac{a_{65}}{a_{37}}$$

Commented by Saham last updated on 14/Feb/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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