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)

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)

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

To find the ratio of the third term to the fourth term in an arithmetic sequence, we can use the given information.

Let's label the first term as a, the second term as b, the third term as c, and the fourth term as d.

Given that the ratio between the first term and the second term is 3/4, we can write the equation:

b = (3/4) * a

Given that the ratio between the second term and the third term is 4/5, we can write the equation:

c = (4/5) * b

To find the ratio of the third term (c) to the fourth term (d), we need to express d in terms of c. We can use the equation from the second term ratio:

c = (4/5) * b

Substitute the value of b using the equation from the first term ratio:

c = (4/5) * [(3/4) * a]

Simplify:

c = (4/5) * (3/4) * a

c = (12/20) * a

c = (3/5) * a

To find the ratio of c to d, we need to express d in terms of c. We can use the formula for the next term in an arithmetic sequence:

d = c + (c - b)

Substitute the values of b and c:

d = c + (c - (3/4) * a)

Simplify:

d = c + c - (3/4) * a

d = 2c - (3/4) * a

To find the ratio of c to d, divide c by d:

(c/d) = (3/5) * a / (2c - (3/4) * a)

To simplify this ratio, multiply the numerator and denominator by 4 and eliminate the fraction:

(c/d) = (3/5) * (4a) / (2(4c) - 3a)

Simplify further:

(c/d) = (12a) / (8c - 3a)

Finally, we can simplify this ratio by factoring out a common factor:

(c/d) = (3a) / (2(4c) - 3a)

(c/d) = (5/6)

So, the ratio of the third term to the fourth term is 5/6.

Now, let's find the ratio of the nth term and the (n+1)th term in the sequence.

We'll express the nth term as a + (n-1)d, and the (n+1)th term as a + nd, where d is the common difference.

To find the ratio, divide the (n+1)th term by the nth term:

[(a + nd) / (a + (n-1)d)] = (a + nd) / (a + (n-1)d)

[(a + nd) / (a + (n-1)d)] = [(a + nd) / (a + nd - d)]

So, the ratio of the nth term and the (n+1)th term in the sequence is (n+2)/(n+3).