Sequential Subtraction

[bk_syed]

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

This method of Plugging in the values is kind of long.

Can anyone explain me the alternate method given?

S2 - S1 = 1 or (20)
S3 - S2 = 2 or (21)
S4 - S3 = 4 or (22)
S5 - S4 = 8 or (23)

S10 - S9 = 28 = 256.


Thanks

[tim]

yeah this is a great alternate approach - basically looking for a pattern in earlier cases. since the problem is looking for the difference between successive values, take a look at what happens when you take successive values at earlier stages of the process. then you can extrapolate to higher values of the sequence..

[hkparikh09]

I had a problem with this question as well from the MGMAT question bank for EIVs.

Is [Sn = 2Sn-1 - 2] a recursive exponential formula? If it is, can we convert to a direct formula?

I tried but was not successful:

S1 = 3 = x (2^n) -2 (where n=1)

5 = x (2^n) (where n=1)

x = 5/2

Direct: Sn = (5/2) (2^n) -2
However, this does not get you the correct values for S9 and S10. Where did I go wrong?

[RonPurewal]

You can't get that kind of thing to work, because that formula only subtracts 2 once (after the whole power is multiplied together).
In the actual sequence at hand, you're subtracting 2 again every single time. There's no way you can get an exponential function to do that.

[RonPurewal]

If you want a formula, then you can notice that this is a "geometric series": 3 + (1 + 2 + 4 + 8 + ...)
if you leave out the "3", then you have a series in which the thingy doubles every time.

on the other hand, that would be an absolutely horrible way to do this problem, for at least three different reasons:

1/ By the time you figure out it's a geometric series, you've already noticed the pattern that solves the problem. So, why waste the time? Ignore the geometric series and just extrapolate the pattern.

2/ The formula is horrible.

3/ To find s10 - s9, you'd have to use the formula two different times and then subtract the results. Ugh.
Especially "ugh" as compared to extrapolating the pattern, in which s10 - s9 will appear immediately as a single value.