In the sequence shown above, a(n) = a(n-1) + k

[Harish Dorai]

a(1), a(2), a(3), ......, a(15).

In the sequence shown above, a(n) = a(n-1) + k, where 2 <= n <= 15 and k is a non-zero constant. How many terms in the sequence are greater than 10?

1) a(1) = 24

2) a(8) = 10

[GMAT 2007]

The answer is (C)

a(n) = a(n-1)+ k

from (1) a(2) = a(1) + k = 24 +k, but we don't what k is, So insufficient, similarly
from (2) a(8) = 10 = a(7) + k, but we don't know what a(7) and k is. So insufficient.

from question stem, we know: -

a(7) = a(6) + k
a(6) = a(5) + k and so on so a(7) = a(2) + 6k.......(3)

from (1) we know a(2) = a(1) + k = 24 +k
from (2) a(8) = a(7) + k so a(7) = 10-k

Putting the values of a(2) and a(7) in (3)
10-k = 24+6k

so k = -2, now no of terms greater than 10 can be calulated.

So (C) is the correct answer

[Harish Dorai]

The answer is actually (B) and of course I couldn't think about this when I took the test ;-)

If a(n) = a(n-1) + k, then a(1), a(2),.....a(15) is an arithmetic progression.

Statement (2) says a(8) = 10

a(8) is the middle term and hence the Median of this series. The number k can be positive or negative. If k is negative, then the numbers a(1), a(2) etc. will be in descending order and if it is positive, then the series will be in ascending order. No matter what, there will be 7 terms which will be greater than 10 (Please note that since k is non-zero, and hence all the terms in the series will be DISTINCT).

It could a(1) to a(7) if k is negative, and a(9) to a(15) if k is positive.

This is definitely a trap kind of questions.

[GMAT 2007]

That's right Harish. Infact, while answering the question, it almost caught my eye, but I was in flow so didn't realize. Thanks for the posting good questions. This one certainly has a trap. :)

GMAT 2007

[Guest]

How is a(8) the middle term? Isn't the middle term of this set the average of a(8) and a(9)?

[RonPurewal]

How is a(8) the middle term? Isn't the middle term of this set the average of a(8) and a(9)?

nope.

most convincing argument: go ahead and list them yourself, and count. there are seven terms before a8, and seven terms after it. therefore, it's the middle term.
if you draw a line between a8 and a9, there will be eight terms to the left of the line and seven terms to the right of the line. therefore, that line is not the middle of the sequence.

--

general facts: (let n stand for the number of terms)

* if you have an even # of terms
in this case, the 'middle term' (median) is the average of the two middle terms, which are term number (n/2) and term number (n/2 + 1).

* if you have an odd # of terms
in this case, the middle term is just one term: term number (n + 1)/2.

hope that helps

[senthilkumar4444]

I think there is something missing in the question. The value for n is not defined.
If it is 15 then answer is 2 that is second option alone enough to answer the question (a(8) = 10) but n is unknown then 1st option also reqd.

So either there is a mistake in the question or answer is both options reqd.

[RonPurewal]

I think there is something missing in the question. The value for n is not defined.

yeah, it's defined.
the prompt specifies that the sequence is strictly limited to 2 < n < 15.

if you didn't realize this, make sure you take note for next time!

[srinivasreddy.c]

though the range of n is specified as 2<= n <= 15, the first statement of the question tells us that a1 is also part of the sequence.

[esledge]

though the range of n is specified as 2<= n <= 15, the first statement of the question tells us that a1 is also part of the sequence.

Yes, you are right, srinivas. The key words are in the question stem: "In the sequence shown above," which was a(1) through a(15), inclusive. I think Ron just intended to say: the terms a(n) are only defined with a recursive function for 2<=n<=15, while a(1) is simply given outright so that a(2) can be calculated from it. However, the entire sequence is 15 terms long.

As he said a few posts back:

How is a(8) the middle term? Isn't the middle term of this set the average of a(8) and a(9)?

nope.

most convincing argument: go ahead and list them yourself, and count. there are seven terms before a8, and seven terms after it. therefore, it's the middle term.

[sudaif]

I thought since function is only defined for values of n given by 2 < n < 15, there were 14 terms in the sequence.
If you don't consider statement 1, why would you assume a(1) is a part of the sequence. A bit confused....

[RonPurewal]

I thought since function is only defined for values of n given by 2 < n < 15, there were 14 terms in the sequence.
If you don't consider statement 1, why would you assume a(1) is a part of the sequence. A bit confused....

two solid reasons.

1) probably the easiest way to figure this out: look at the list they give you in the prompt (no joke)
this list starts with a(1), not with a(2). therefore, the first term in the sequence is a(1), not a(2).

2) look at the recursive formula. the problem states that the formula is good for 2 < n < 15 -- i.e., you should be able to plug in each of the integers from 2 to 15 into that formula, and get something meaningful.
if you plug the integer n = 2 into that formula, the formula requires you to use a(n-1) = a(1).

[jp.jprasanna]

I thought since function is only defined for values of n given by 2 < n < 15, there were 14 terms in the sequence.
If you don't consider statement 1, why would you assume a(1) is a part of the sequence. A bit confused....

two solid reasons.

1) probably the easiest way to figure this out: look at the list they give you in the prompt (no joke)
this list starts with a(1), not with a(2). therefore, the first term in the sequence is a(1), not a(2).

2) look at the recursive formula. the problem states that the formula is good for 2 < n < 15 -- i.e., you should be able to plug in each of the integers from 2 to 15 into that formula, and get something meaningful.
if you plug the integer n = 2 into that formula, the formula requires you to use a(n-1) = a(1).

Hi Ron - Sorry for digging this post after a very long time... I have 2 questions here....

1. even thought we have a(1) = 24 we still wont be able to calculate the sequence right since we need a(0) as well right to get the value of K isn't it? So no use with Statement 1 - Correct? Immaterial what the sequence is or how many terms are there in the sequence etc. I cant do anything with A - Please correct me if im worng here!?


2. question stem reads " In the sequence shown above, a(n) = a(n-1) + k, where 2 <= n <= 15"

So the above statement can be interpreted as the formula - a(n) = a(n-1) + k - holds good when n is between 2 to 15 (inclusive)

so we don't actually know how many terms the sequence has right i.e the sequnce could me made of 15 or even 10...? then the ans B woudn't be sufficient isn't it?

3. While estimating statement B we should / must ignore statement A... only clue for the sequence starting with a(1) is from stat 1 isn't it? otherwise we can guess the sequence might start from 2 as stated in the question stem - 2 <= n <= 15

Im sorry to bother you with so many question but please help...

Cheers
JP

[RonPurewal]

jp, the very first thing in the problem statement is a list of the whole sequence: a(1), a(2), ..., a(15).
you can think of this as an announcement that the sequence has fifteen elements in it.

1. even thought we have a(1) = 24 we still wont be able to calculate the sequence right since we need a(0) as well right to get the value of K isn't it? So no use with Statement 1 - Correct? Immaterial what the sequence is or how many terms are there in the sequence etc. I cant do anything with A - Please correct me if im worng here!?

there's no such thing as a(0). the first term in the sequence is a(1).

this is why the recursive rule starts at n = 2: a(1) is the first term of the sequence, so it's not generated recursively. the first term that's generated by the formula a(n) = a(n-1) + k is the second term, for which n = 2.

so we don't actually know how many terms the sequence has right i.e the sequnce could me made of 15 or even 10...? then the ans B woudn't be sufficient isn't it?

the beginning of the problem statement makes it clear that the sequence runs from a(1) to a(15), no more, no less.

3. While estimating statement B we should / must ignore statement A... only clue for the sequence starting with a(1) is from stat 1 isn't it? otherwise we can guess the sequence might start from 2 as stated in the question stem - 2 <= n <= 15

the sequence runs from a(1) to a(15).

[career.sumit]

what is the OA ?

I got the answer as B, am i right ?