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Word Problem - run 1 mile on Monday











































Geometric Progression

If you run 1 mile on Monday, and then double it everyday, how many miles would you run in 5 days?

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Oct 12, 2012
Geometric Progression
by: Staff


Answer:

Part I

The sequence of miles run you have described is called a geometric progression (or geometric sequence)

A geometric sequence has the (general) form:

xn = x1 * (r)(n - 1)


xn = x with a subscript of n (this is the nth term in the sequence)

x1 = x with a subscript of 1 (this is the 1st term in the sequence)

n = number of terms

r = the common ratio

r, the common ratio, can be calculated as follows:
rn = xn / xn-1

(n must be greater than 1)

rn = r with a subscript of n (this is the common ratio)

xn = x with a subscript of n (this is the nth term in the sequence)

xn-1 = x with a subscript of n-1 (this is the n-1 term in the sequence)


r2 = x2/ x1

r2 = (2)/ (1)

The common ratio, r = 2

The first five terms in the geometric sequence:
       

xn = x1 * (r)(n - 1)

xn = 1 * (2)(n - 1)


If n = 1 (the first day), the first term is 1 mile

xn = 1 * (2)(n - 1)

x1 = 1 * (2)(1 - 1)

x1 = 1 * (2)0

x1 = 1 * 1

x1 = 1 mile



If n = 2 (the second day)

xn = 1 * (2)(n - 1)

x2 = 1 * (2)(2 - 1)

x2 = 1 * (2)1

x2 = 1 * 2

x2 = 2 miles



If n = 3 (the third day)

xn = 1 * (2)(n - 1)

x3 = 1 * (2)(3 - 1)

x3 = 1 * (2)2

x3 = 1 * 4

x3 = 4 miles



If n = 4 (the fourth day)

xn = 1 * (2)(n - 1)

x4 = 1 * (2)(4 - 1)

x4 = 1 * (2)3

x4 = 1 * 8

x4 = 8 miles



If n = 5 (the fifth day)

xn = 1 * (2)(n - 1)

x5 = 1 * (2)(5 - 1)

x5 = 1 * (2)4

x5 = 1 * 16

x5 = 16 miles


---------------------------------------------------------

Oct 12, 2012
Geometric Progression
by: Staff


---------------------------------------------------------
Part II

The sum of a geometric sequence is given by the formula:
         x1(1 -  rn)

Sn = ---------------
(1 - r)



The sum of a miles run for all five days is:
         1 * (1 -  25)

S5 = ---------------
(1 - 2)

1 * (-31)
S5 = ---------------
(-1)

S5 = 31




Final Answer:

                 Distance Run the fifth day = 16 miles

                 Sum of Distance Run for all 5 days = 31 miles



Thanks for writing.

Staff
www.solving-math-problems.com



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