DUSUN the removed cards is k, then k-20-
Let a, b, c be positive integers such that - is an integer. If a, b, c are in geometric
20131
progression and the arithmetic mean of a, bc is b + 2. then the value of
ata-14
Answers
Answer:
This may look so big
but pls try this
Step-by-step explanation:
This question requires patience. Because I remember the first time, I tried this one, it took me a while.
So let's just gather informations and then club them together to find out what we can.
We know
b/a is an integer
a, b and c are in GP.
Arithmetic mean of the three numbers is b+2.
And that (a^2 + a - 14)/(a - 1) has got something to do with it.
We'll start with the second point because we can't work with our first one yet.
Second point tells us, that since these numbers, a, b and c are in GP, we can also write them as,
b/r, b, br where r is our common ratio. Because they are still in GP, so there shouldn't be any problem.
Next we will start with our third point. Their Arithmetic mean is b + 2.
Which implies that,
b + 2 = (a + b + c) /3
And therefore we have our next equation,
2b + 6 = a + c.
But what did we take a as? b/r
And c? br
Therefore we get,
2b + 6 = ( b/r ) + br
And from our first point, we know that b/a is an integer,
b/a = b/(b/r) = r is an integer.
Now mind, b should be an integer. This is an important point.
6 = ( b/r ) + br - 2b, taking b common, we have,
6 = b ( r + (1/r) - 2) - - (i)
Do you see it?
r + (1/r) - 2 = ( root(r) - (1/root(r) )^2. Think about it and write it down if you're confused. Expand the whole thing, and you'll see it.
So,
6 = b ( root(r) - 1/(root(r) )^2 - - (ii)
Now we can try to think about either of the two equations, (i) or (ii).
I'll be using (i) because it becomes easier to think.
If I try r = 1, it wouldn't make any sense, because,
b * 0 = 0, not 6. So let's try r =2 next, and this time, Voila!!
b/2 = 6, hence,
b = 12, and it's an integer, which is what we wanted. You could have a try at r = 3, r = 4 but in all of those cases, b would not be an integer and our cases would be invalid.
Hence, we have found out that r = 2, and b = 12
Since, our a = b/r = 6. And we will put it in our required equation, we get
(a^2 + a - 14)/(a - 1) = (36 +6 - 14)/5 = 28/5. Our answer.