Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

SP # 1

If p, q, r are in Arithmetic Progression and x, y, z are in Geometric Progression, show that

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

Let:

p=p

q=p+a

r=p+2a

x=x

y=bx

z=b^2x

Then:

*Last edited by krassi_holmz (2006-02-27 03:05:26)*

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

krassi_holmz, although I don't see any serious mistake in the way you started, I am not fully convinced with the proof. I shall wait for a few more days before posing the solution.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

p,q and r are in Arithmetic prgression, so

q=p+a

r=p+2a, because of the arithmetic progression propeties.

Same for the x,y,z:

y=bx

z=b^2x

Next is just simple arithmetic reduction:

Where's my mistake?

*Last edited by krassi_holmz (2006-02-27 17:46:07)*

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

That's better.

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

SP # 2

The sum of the digits of a three digit number is 12. The digits are in Arithmetic Progression. If the digits are reversed, then the number is diminished by 396. Find the number.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

642?

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

I want MORE!!!

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

Here you get!

SP# 3

The sum of an infinite series in Geometric Progression is 57 and sum of their cubes is 9747. Find the series.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

SP # 4

A ball is dropped from a height of 6m and on each bounce it rebounces to 2/3 of its previous height. How far does the ball travel till it stops bouncing?

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

SP #4: the ball is dropped, so it doesn't travel anywhere.

But seriously, by traveled, do you mean both positive and negative changes in height? In other words, do we count the ball going up and down?

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

If we count this we get the sum :

I may be wrong.

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

This is how the problem is solved in a different way.

1. The distance covered in the downward path is an infinite Geometric series with a=6m, r=2/3.

Therefore, S[sub]n=[6/(1-2/3)]=6/(1/3)=18m

2. The distance covered in the upward path is an infinte Geometric series with a=4m, r=2/3.

S[sub]n=[4/(1-2/3)]=4/(1/3)=12m

Total distance = 18m + 12m = 30m.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Ricky wrote:

SP #4: the ball is dropped, so it doesn't travel anywhere.

If you're being picky like that, then technically it travels 6m.

Why did the vector cross the road?

It wanted to be normal.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

Einstein would say:

It depends on it's speed.

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

SP # 5

The first term of a Geometric Progression is 64 and the average of the first and the fourth terms is 140. Find the common ratio 'r'.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

*Last edited by Ricky (2006-03-05 17:06:02)*

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

**Well done. Ricky!**

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

SP # 6

A man borrows $5,115 to be repaid in 10 monthly instalments. If each instalment is double the value of the last, find the value of the first and the last instalment.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Why did the vector cross the road?

It wanted to be normal.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,953

**You are correct, mathsyperson! Well done! **

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

SP#3:

q=57(1-a)

Solving

a=2/3 or a=3/2;

Then q=19 or q=-57/2

But when q=-57/2 the sum is negative, so:

So the answer is:

a=2/3;q=19

IPBLE: Increasing Performance By Lowering Expectations.

Offline