Question

The ratio between the first term and the second term in an arithmetic sequence is 3/4. The ratio between the second term and the third term is 4/5.

a. Calculate the ratio of the third term to the fourth term. (Answer is 5/6)
b. Find the ratio of the nth and the nth + 1 term in the sequence. (Answer is n+2/n+3)

Answer 1

a1/a2 = 3/4
a2/a3 = 4/5
huh - by just looking at this, you can see that the sequence is 3,4,5,6,...
so, a3/a4 = 5/6
of course, 6,8,10,12,... will also work. To justify this easy guess, just plug and chug:

a/(a+d) = 3/4
(a+d)/(a+2d) = 4/5

4a = 3a+3d
5(a+d) = 4(a+2d)

a = 3d
so, it does not matter what d is. the 1st term is just 3d. Now, to find
an/a(n+1) = (a+(n-1)d)/(a+nd) = (a+nd-1)/(a+nd)
= (3d+nd-1)/(3d+nd)
So, picking the simplest sequence, we have d=1
= (n+2)/(n+3)
If d=2, the ratio is (2n+5)/(2n+6)

Answer 2

Oops. Did you catch my typo? Look again...