A tour bus travels 400 miles on the first day of a trip. Each day after that the bus travels half. The total is 775 miles. How many days does the trip take?

Question

Im a bit confused on this question

Answer 1

this is a geometric sequence

400, 200, 100, ... where the common ratio is 1/2

you can use the formula for the sum of a geometric sequence or use simple arithmetic

using the formula

S=a(r^n-1)/(r-1) where a=first term, r=common ratio, and n=number of terms

775=400([1/2]^n-1)/([1/2]-1)

775=400([1/2]^n-1)/(-1/2)

775=-800([1/2]^n-1)

775/-800=([1/2]^n-1)

-31/32=[1/2]^n-1

1-31/32=[1/2]^n

1/32=[1/2]^n

[1/2]^n=1^n/2^n

1^n=1 for all n

now we have 1/2^n

32=2^5

1/(2^5)=1/(2^n)

since the numerators are equal the denominators are equal

2^5=2^n

therefore n=5

the trip takes 5 days

 

using simple arithmetic

400+200=600

400+200+100=700

400+200+100+50=750

400+200+100+50+25=775

there are 5 terms so it takes 5 days for the trip