# 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?

### Comments for Word Problem - run 1 mile on Monday

 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