Sequential Subtraction

[Silky35]

Can someone explain this to me?

Sequence S is defined as Sn = 2Sn-1 - 2. If S1 = 3, then S10 - S9 =

[SILKY35]

the answer is below. I just don't see it because I am an idiot

S1 = 3 S2 = 2(3) - 2 = 4 S3 = 2(4) - 2 = 6
S4 = 2(6) - 2 = 10 S5 = 2(10) - 2 = 18 S6 = 2(18) - 2 = 34
S7 = 2(34) - 2 = 66 S8 = 2(66) - 2 = 130 S9 = 2(130) - 2 = 258
S10 = 2(258) - 2 = 514

S10 - S9 = 514 - 258 = 256

[StaceyKoprince]

This is one of those annoying "recursive" sequence problems. You have to keep plugging in to find the next term in the sequence; you can't just calculate S10 and S9 outright. (Note: on the test, they don't typically make you do 10 calculations like this. Instead, if it seems you would have to do that, there's usually a pattern you can use to find the subsequent numbers.

They tell us S-sub-1 equals 3. To get S-sub-2, I have to plug in the value for S-sub-1:

S-sub-n = 2*S-sub-(n-1) - 2

I'm looking for S-sub-2, so substitute 2 in for n:
S-sub-2 = 2*S-sub(2-1) - 2
S-sub-2 = 2*S-sub-1 - 2
You know S-sub-w (it's 3!), so plug it in:
S-sub-2 = 2*3 - 2
S-sub-2 = 6-2 = 4

Now I want to find S-sub-3 and I have to do the whole thing all over again, this time plugging in 3 for n:
S-sub-3 = 2*S-sub-(3-1) - 2
S-sub-3 = 2*S-sub-2 - 2
again, you know S-sub-2, so plug it in:
S-sub-3 = 2*4 - 2
S-sub-3 = 8-2 = 6

If you do the above work again (try it) for S-sub-4, you'll get that S-sub-4 = 10. Do it again for S-sub-5, and you'll get 18.


Here's your sequence so far: 3, 4, 6, 10, 18
What's happening each time? The first time (between 3 and 4), we added 1. Then we added two (4 to 6). Then we added 4. Then we added 8. I wonder whether we'll add 16 next time? Let's see: do the math for S-sub-6 and see what you get. (It's 34.) Hmm... 18 + 16 = 34!

So basically, we're adding a multiple of 2 each time: 1, then 2, 4, 8, 16, ... I could use that pattern to figure out the values for S-sub-9 and S-sub-10 more quickly (but don't get too freaked out - this is a more complicated pattern than what you'd be expected to do on the test).

So S-sub-7 should be 34 + (16*2) = 34+32 = 66.
S-sub-8 = 66 + (32*2) = 66+64 = 130
S-sub-9 = 130 + 64*2 = 130+128 = 258
S-sub-10 = 258 + 128*2 = 258 + 256 = 514

And then do that last bit: 514-258 to get your answer. There are some ways you could simplify this one even more but, again, this is more complicated than what the test would expect you to do. This kind of stuff is just to give you practice with carrying out some of these calculations in case you need to.