Ratios are easy stuff, right?

Algebra Level 3

The ratio of the fifth and the third term of an arithmetic progression is $2:5$ . Find the ratio of the $15^\text{th}$ and the $7^\text{th}$ term.

4:3

13:1

5:7

12:7

1 solution

Abc Xyz
Mar 9, 2016

Let us assume that,

Third term = a+ 2d and Fifth term = a + 4d

Therefore their ratio is $\frac{a+4d}{a+2d} = \frac{2}{5}$

By cross multiplying and solving we get a= $\frac{-16}{3}d$

Now let's take

Fifteenth term = a + 14d Seventh term = a + 6d

Lets substitute and take their ratio.

We get $\frac{26d}{3}/\frac{2d}{3}$

By cancelling d and the 3 we get $\frac{26}{2}$

Which is $\boxed{13 : 1}$ .

*Note: the slash in 3rd last line means divide..

I'm curious as to how you got a = -16/3 when there are two variables in your equation.

When cross multiplied, I get 5a + 10d = 2a + 8d or 2d + 3a = 0. Or even d/a = -3/2

The general solution is d = 3t and a = -2t.

So the arithmetic sequence is a_n = -2t + (n - 1)(3t).

a 15 = 42t - 2t = 40t a 7 = 18t - 2t = 16t

a 7/a 15 = 16t/40t = 2/5

So they have the same ratio.

Ashley Reavis - 5 years, 3 months ago

Thank u for pointing out my mistake......I have edited the answer. Sorry for my carelessness...

abc xyz - 5 years, 3 months ago