+1207 Bob wants to read an 80 page book. On the first day he reads 5 pages, and for each subsequent day he decides to read \(\frac{1}{2}\) page more than the previous day. How many days does Bob need in order to finish the book?
This is an arithemetic sequence. Use the arithmetic sequence formula to solve for N:
Solve for N:
N/2*[2*5 + 1/2*(N - 1) ]= 80, solve for N
1/2 N ((N - 1)/2 + 10) = 80
Multiply both sides by 2:
N ((N - 1)/2 + 10) = 160
Expand out terms of the left hand side:
N^2/2 + (19 N)/2 = 160
Multiply both sides by 2:
N^2 + 19 N = 320
Add 361/4 to both sides:
N^2 + 19 N + 361/4 = 1641/4
Write the left hand side as a square:
(N + 19/2)^2 = 1641/4
Take the square root of both sides:
N + 19/2 = sqrt(1641)/2 or N + 19/2 = -sqrt(1641)/2
Subtract 19/2 from both sides:
N = sqrt(1641)/2 - 19/2 or N + 19/2 = -sqrt(1641)/2
Subtract 19/2 from both sides:
N = sqrt(1641)/2 - 19/2 = 10.75, or about ~11 days to read the book
+37146 For a geometric series Sn = a1 (1-rn) / (1-r)
The common ratio is 1.5 (1.5 more pages each day) a1 = 5
80 = 5 (1- 1.5n) / (1-1.5)
-40 = 5 (1-1.5n )
-8 = (1-1.5n)
-9 = -1.5n
9 = 1.5n Take log of both sides
log9/log1.5 = n = 5.41 days ~~~ 6 days to read the 80 page book.
+129852 The arthmetic "formula" for the number of pages read on Day N is given by :
5 + (N-1)(.5) =
5 + .5N - .5 =
4.5 + .5N
The first term = 5
And the last term is 4.5 + .5N
So....we want to solve this :
(N/2) [ first term + last term ] = 80
(N/2)[ 5 + 4.5 + .5N ] = 80
(N/2) [ 9.5 + .5N ] = 80 mutiply through by 2
N [ 9.5 + .5N ] = 160
.5N^2 + 9.5N - 160 = 0 multiply through by 2 again
N^2 + 19N - 320 = 0
The graph here (letting N = x) shows two possible solutions : https://www.desmos.com/calculator/ze8gobvp4n
Taking the positive one...it will take about 10.75 days = 11 days
