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: 15,389

SP # 1

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

Character is who you are when no one is looking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

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: 15,389

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.

Character is who you are when no one is looking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

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: 15,389

Character is who you are when no one is looking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

That's better.

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 15,389

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.

Character is who you are when no one is looking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

642?

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 15,389

Character is who you are when no one is looking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

I want MORE!!!

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 15,389

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.

Character is who you are when no one is looking.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 15,389

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?

Character is who you are when no one is looking.

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: 1,908

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: 15,389

Character is who you are when no one is looking.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 15,389

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.

Character is who you are when no one is looking.

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: 1,908

Einstein would say:

It depends on it's speed.

IPBLE: Increasing Performance By Lowering Expectations.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 15,389

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'.

Character is who you are when no one is looking.

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: 15,389

**Well done. Ricky!**

Character is who you are when no one is looking.

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 15,389

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.

Character is who you are when no one is looking.

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: 15,389

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

Character is who you are when no one is looking.

Offline

**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

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