On-a-path-there-was-an-odd-number-of-rocks-with-a-distance-of-10m-between-them-we-want-to-put-them-all-where-the-middle-one-is-if-we-start-to-collect-from-one-of-the-ends-and-when-we-finish-we-

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Question Number 182704 by HeferH last updated on 13/Dec/22

$$Onapaththerewasanoddnumberofrocks$$$$withadistanceof10mbetweenthem,we$$$$wanttoputthemallwherethemiddleoneis,$$$$ifwestarttocollectfromoneoftheendsand$$$$whenwefinishwehavewalked3kmintotal,$$$$howmanyrocksarethere?$$

Answered by TheSupreme last updated on 13/Dec/22

$$ifn=1S_1=0$$$$ifn=3S_3=3\ast 10$$$$ifn=5S_5=S_3+10+2\ast 3\ast 10$$$$S_{2n+1}=S_{2n-1}+10\ast (n-1)+3\ast n\ast 10$$$$S_{2n+1}=\underset{i=1}{\sum ^n}10(i-1)+30\ast i=\underset{i=1}{\sum ^n}40i-10$$$$S_{2n+1}=40\frac{n(n+1)}{2}-10n$$$$3000=20n^2+10n=10n(2n+1)$$$$n=12$$$$$$

Answered by mr W last updated on 13/Dec/22

$$saythereare2n+1rockswithdistanced$$$$betweenthem.$$$$totaldistancetowalk:$$$$L=4(d+2d+3d+\dots +nd)-nd=n(2n+1)d$$$$3000=n(2n+1)10$$$$2n^2+n-300=0$$$$\Rightarrow n=12$$$$i.e.thereare2\times 12+1=25rocks$$

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