how many arithmetic progressions of six increasing terms include the terms 15 and 20

Question

this is so hard please help because i dont get it

Answer 1

This problem is basically asking you to start at the number 15 and figure out how many ways there are to get to the number 20 with 5 or fewer equal steps. (It's 5 and not 6 because 15 is the first term in the progression, so we've already used one of the 6 terms.)

 

 

 

I can do it easily in one step:

 

15, 20

 

That gives your progression a common difference of 5, so its 6 terms are:

 

15, 20, 25, 30, 35, 40

 

But if we just want to include 15 and 20, we could also have:

 

10, 15, 20, 25, 30, 35

5, 10, 15, 20, 25, 30

0, 5, 10, 15, 20, 25

-5, 0, 5, 10, 15, 20

 

That's a total of five progressions using a common difference of 5.

 

 

 

I can also get from 15 to 20 in two steps:

 

15, 17.5, 20

 

That progression has a common difference of 2.5, so its six terms are:

 

15, 17.5, 20, 22.5, 25, 27.5

 

But again, if we just want to include 15 and 20, we can also have:

 

12.5, 15, 17.5, 20, 22.5, 25

10, 12.5, 15, 17.5, 20, 22.5

7.5, 10, 12.5, 15, 17.5 20

 

That's a total of four progressions using a common difference of 2.5.

 

 

 

I can also get from 15 to 20 using three steps:

 

15, 16 ²/₃, 18 ¹/₃, 20, 21 ²/₃, 23 ¹/₃

 

That progression has a common difference of ⁵/₃.

 

We can also have:

 

13 ¹/₃, 15, 16 ²/₃, 18 ¹/₃, 20, 21 ²/₃

11 ²/₃, 13 ¹/₃, 15, 16 ²/₃, 18 ¹/₃, 20

 

That's a total of three progressions using a common difference of ⁵/₃. I'm beginning to sense a pattern...

 

 

 

We can get from 15 to 20 in four steps:

 

15, 16 ¹/₄, 17 ¹/₂, 18 ³/₄, 20, 21 ¹/₄

 

That's a common difference of ⁵/₄, which will also give us:

 

13 ³/₄, 15, 16 ¹/₄, 17 ¹/₂, 18 ³/₄, 20

 

That's two progressions with a common difference of ⁵/₄.

 

 

 

And finally, from 15 to 20 in five steps:

 

15, 16, 17, 18, 19, 20

 

That's one progression with a common difference of 1.

 

 

 

There aren't any more possible progressions. Anything with a smaller common difference will take more than six terms, and anything with another common difference will skip over either 15 or 20. So, we've got a grand total of 5 + 4 + 3 + 2 + 1 = 15 arithmetic progressions of six increasing terms that include the terms 15 and 20.