Sequences

Question

Hello , 1st of all , i don't have the answer choices , BUT ITS VERY Important Question ...

Q: 
if a1= 1 and a2=100 and every an above a2 is equal to the average of the two preceding a , what is a10 ?

4gmatmumbai · Answer

Hi,

a3 = (a1+a2)/2

a4 = (a1 + 3a2)/4

a5 = (3a1 + 5a2) / 8

a6 = (5a1 + 11a2) / 16

a7 = (11a1 + 21a2) / 32

a8 = (21a1 + 43a2) / 64

The pattern is that the coefficient of a2 in the previous term becomes the coefficient of a1 in the current term. The coefficient of a2 is twice of the coefficient of a1 + 1.

Similarly, a10 = (85a1 + 171a2) / 256

It boils down to 67.129. I suppose the answer choices will be more helpful (or) will be in terms of an expression rather than a decimal number.

Hope this helps.

Thanks.

Pavle · Answer

Very good explanation Kudos

IanStewart · Answer

You really *need* the answer choices here to know what approach to take. If the question asks for an exact decimal value, it's a time consuming question. If this were on the GMAT, it's more likely we could get away with some kind of approximation. When you average two numbers x and y, you are just finding the number halfway between x and y. So you could draw a number line and zero in pretty quickly on the rough location of a_10.:

-1-------------a3---a5--a6--a4--------100

I've drawn the first six terms, but you can see that a_10 is confined to a pretty narrow range on the number line, and if you simply calculate a_3 and a_4, you can narrow down the possibilities considerably. If the answer choices did not ask for an exact value, this kind of approach would probably be all that's necessary.

And  why  is this question "VERY important"?

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