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# How do u apply this to a arithmetic series? And what would be the answer if it was?

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Peter Pedals rode his bike a total of 500 miles in five days. Each day he rode 10 more miles than he had ridden on the previous day. How many miles did Peter ride on just the fifth day?

Aug 21, 2019

#1
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$$miles_1 = M\\ miles_k = miles_1 + 10(k-1),~k=1,2,\dots 5\\ \sum \limits_{k=1}^5 miles_k = \sum \limits_{k=1}^5 M+10(k-1) = 500\\ 5M + 10\sum \limits_{k=0}^4 k = 500\\ 5M + 10\dfrac{(4)(4+1)}{2} = 500\\ 5M + 100 = 500\\ M=80\\ miles_5 = 80 + 10(4) = 120~miles$$

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Aug 21, 2019
#2
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Call the distance he bikes the first day  =  D

The distance he bikes the  ffth day  =  D + 4(10)  = D + 40

So

Sum  =   [ Distance on first day + Distance on fifth day] * (number of days / 2)

500  =  [ D  + D + 40 ] *  (5/2)

500 = [ 2D + 40 ]  * (5/2)     multiply both sides by 2/5

200  =   2D + 40     subtract 40 from each side

160  =  2D    divide both sides by 2

80 = D

So....on the fifth day he bikes  80 + 40  =  120 miles   Aug 21, 2019