Question

how many four digit numbers have the property that their digits taken from left to right form an arithmetic or geometric progression​

Answer 1

Answer:

Step-by-step explanation:

Let us count. The list of geometric progressions is short, though not as short as I first believed! There are the obvious 1,2,4

, and 2,4,8, and 1,3,9, and their reverses. Then there is the less obvious 4,6,9 and its reverse, which I had missed. And of course a,a,a where a is any of 1 to 9. These last also happen to be arithmetic progressions. What about "common ratio" 0

? We will go along with Wikipedia's definition and not allow that.

Now let's count the arithmetic progressions. There are the 9

with common difference 0

.

There are 7

increasing ones with common difference 1, and 8 decreasing ones, since 0

can be the final digit in that case.

There are 5

increasing ones with common difference 2, and 6

decreasing ones.

There are 3

increasing ones with common difference 3, and 4

decreasing ones.

There is 1

increasing one with common difference 4, and there are 2

decreasing ones.

If we decide to forget about the sequences a,a,a

we get a count of 44, for we listed 6 geometric progressions and 36 arithmetic progressions. Adding in the sequences a,a,a, which we definitely should, since they are indeed in both categories, gives us 53.